# C3: Functions. Learning objectives

Size: px
Start display at page:

Transcription

1 CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the term function be able to find the range of a function be able to form composite functions understand the condition for an inverse function to eist.. Notation In the first chapter of C ou were introduced to function notation. It is rather like having a machine into which numbers are fed, and for each value input, the machine determines the output value. The function f which squares the number input and then adds 3 to the result can be represented b f() 2 3 so that f() 3 4 and f( 4) The epression f() is sometimes called the image of. The letter f is frequentl used to represent a function, since it is the first letter of the word function, but it is quite in order to use an other letter instead. However, when ou have two different functions, it is usual to call the first one f and the second one g, and so on. Input function machine utput An alternative notation for f, often used in universit tets, is f: 2 3. However this notation will not be used in the AQA eaminations. Worked eample. The functions f and g are defined for all real values of and are such that f() 2 4 and g() 4. (a) Find f( 3) and g(0.3). (b) Find the two values of for which f() g().

2 2 C3: Functions Solution (a) f( 3) ( 3) g(0.3) (4 0.3) (b) Since f() g() ou can write Therefore Factorising gives ( 5)( ) 0, so that 5 or. Hence the two values of for which f() g() are 5 and. EXERCISE A The function f is defined for all real values of b f() 2 3. Find the values of: (a) f( ), (b) f(3). 2 The function g is defined for all real values of b g() (2) 3. Find the values of: (a) g( ), (b) g(3). 3 Given that s() 3 2sin, find: (a) s(0 ), (b) s(90 ), (c) s(30 ), (d) s(270 ). 4 Given that t() 4 tan, find: (a) t(0 ), (b) t(45 ), (c) t(80 ), (d) t(35 ). 5 The functions f and g are defined for all real values of and are such that f() 3 5 and g() 4. (a) Find f( ) and g(2). (b) Find the value of for which f() g(). 6 The functions f and g are defined for all real values of and are such that f() 2 2 and g() 5 2. (a) Find f( 3) and g( 5). (b) Find the two values of for which f() g(). 7 Given that f() , evaluate: (a) f(0), (b) f(), (c) f(2), (d) f( ), (e) f( 2). 8 Given that g() ( 3) 3, evaluate: (a) g(0), (b) g(), (c) g(2), (d) g( ), (e) g( 2), (f) g( 4), (g) g( 2.99), (h) g(b), (i) g(a 3). 9 Given that f() (2 5), find the eact values of: (a) f(0), (b) f(), (c) f(2), (d) f( ), (e) f( 2).

3 C3: Functions 3.2 Mapping diagram Instead of finding a single value of f(), imagine that each number in the set { 2, 0,, 3, 4} is input in turn to a function machine. The corresponding output values could be represented as a mapping diagram as shown in the diagram. Can ou recognise what the mapping is actuall doing? Each number is cubed and then one is added to the result. It doesn t matter that no elements are mapped onto 74 and 3. Because each element of the first set is mapped to eactl one element of the second set, we sa the mapping is one-one. The set of input values is called the domain. So for this mapping the domain is the set { 2, 0,, 3, 4}. If ou map from the domain using arrows, the set of values where the arrows map onto is called the range. Here, the range is the set { 7,, 2, 28, 65}. Consider a second mapping diagram as shown. The larger set on the right that contains the range is called the codomain so that, in this eample, the codomain is the set { 3, 7,, 2, 28, 65, 74}. The term codomain will not be used in eamination questions Notice that both of the numbers 3 and 5 are mapped onto 2. Also the numbers 0 and 2 are mapped onto 3. This time the domain is the set { 5, 3, 2, 0, 3} and the range is the set { 3, 0, 2}. In this case more than one element of the domain maps onto the same element in the range. The mapping is man-one. When a mapping is one-one or man-one it is called a function. It is usuall represented b a single letter such as f, g, or h, etc. The set of numbers for which a function is defined is called the domain. A mapping such as illustrated below is one-man and cannot represent a function

4 4 C3: Functions A function f consists of two things: a defining rule such as f() 2 3; its domain. The set of values the function can take for a given domain is called the range. Worked eample.2 The function h has domain { 2,, 0, 3, 7} and is defined b h() ( 3) 2 2. Find the range of h. Solution h( 2) ( 2 3) h( ) ( 3) h(0) (0 3) h(3) (3 3) h(7) (7 3) The range of h is {2,, 8, 27} Because h( ) and h(7) give the same value, we onl write the value 8 once in the range..3 Functions with continuous intervals as domains It is more common for a function to have an interval of values as its domain rather than the domain consisting of just a finite set of values such as { 2, 0,, 3}. Suppose the function f is defined for the domain 2 3 b f() 3 2. The graph of 3 2 is a straight line. The section of the line ou are restricted to is where 2 3, since this is the domain of the function f. Since f( 2) and f(3) 9 2, the onl part of the line to be considered is the section between the two points with coordinates ( 2, 4) and (3, ). (3, ) The graph of f() is drawn opposite. 4 ( 2, 4) It is a good idea to put blobs at the end points to remind ou that these are values the function can actuall take.

5 C3: Functions 5 The possible values that can take are therefore 4. This defines the range of the function. The range of f can be written as 4 or 4 f(). When the domain of f is a continuous interval, the range can be found b considering the graph of f(). The range consists of the possible values that can take. The range of f is written as an inequalit involving f(). Worked eample.3 The function g has domain 2 and is defined b g() 2. (a) Sketch the graph of g(). (b) Find the range of g. Solution (a) The quadratic graph 2 has a minimum point at (0, ). It is useful to evaluate the function at the end points of the domain. Here g( ) 0 and g(2) 4 3. The section of the parabola required is sketched below and the end points are indicated b small blobs. 3 (2, 3) The graph of g() is shown opposite. (, 0) (b) You need to consider more than the two end points when considering the range of values that can take. Notice that the graph comes down as low as. The greatest value can take is 3, as given b the righthand etremit of the graph. Hence the range of g is given b g() 3. EXERCISE B The function f has domain { 2,, 0,, 2, 3} and is defined b f() 2 2. Find the range of f. 2 The function g has domain { 2,, 0,, 2} and is defined b g() (2 ) 3 7. Find the range of g.

6 6 C3: Functions 3 The function h is defined for 3 b h() 3. Find the range of h. 4 The function f is defined for 2 b f() 2. (a) Sketch the graph of f(). (b) Find the range of f. 5 The function g is defined for 4 b g() 0 2. (a) Sketch the graph of g(). (b) Find the range of g. 6 The function f is defined for all real values of b f() ( 2)( 2). (a) (i) Find the coordinates of the points where the graph of f() cuts the coordinate aes. (ii) Sketch the graph of f(). (b) State the range of f. [A].4 Further eamples involving domain and range Sometimes the domain is defined in such a wa as to eclude the end points of the interval. For instance, the domain ma be of the form 3, or perhaps something like 2. Worked eample.4 The function g with domain 2 is defined b g() 3 4. Sketch the graph of g() and find the range of g. Solution 4 (2, 4) You can use a graphics calculator or recognise that the basic graph of 3 has been translated through 0. 4 (, 5) 5 You need to find the smallest and greatest values of g() from the graph. g( ) ( ) g(2) (even though 2 is not in the domain) It is important to sketch the graph onl for the domain indicated. There is a difference between the graph of 3 4, which eists for all real values of, and the graph of g() which is sketched here.

7 C3: Functions 7 Since the function is not defined when 2 it is a good idea to represent this in some special wa. This is usuall done b drawing a small circle to remind ou that the point (2, 4) is not actuall included in the graph. The graph shows that can take all values between 5 and 4. The function can take the value 5 but not the value 4. Hence, the range of g is given b 5 g() 4. Worked eample.5 The function f is defined b f() 4 2, graph of f() and find the range of f. 2. Sketch the Solution It is likel that our first attempt to obtain a sketch will produce something like the one opposite, particularl if ou are using a graphics calculator. However, ou need to restrict the set of values so that onl the part of the graph for > 2 is drawn. The graph of f() for 2 is shown below. Domains ma be defined in this wa in the C3 eamination. Notice that the word domain is not actuall mentioned (2, 0) Although the value 2 is not actuall in the domain ou need to find f(2) As gets larger than 2 ou can see that f() decreases continuousl from the value 0. The range is therefore given b f() 0. EXERCISE C The function g with domain 4 is defined b g() 7. Sketch the graph of g() and find the range of g. You ma wish to use a graphics calculator to help ou with our sketches. 2 The function f with domain 3 is defined b f() 2. Sketch the graph of f() and find the range of f. 3 The function q with domain 2 is defined b q() 3. Sketch the graph of q() and find the range of q.

8 8 C3: Functions 4 The function g is defined b g() 3 5,. Sketch the graph of g() and find the range of g. 5 The function f with domain 2 is defined b f() 3 5. Sketch the graph of f() and find the range of f. 6 The function h with domain 2 3 is defined b h() 2 3. Sketch the graph of h() and find the range of h. 7 The function f with domain is defined b f() 3. Sketch the graph of f() and find the range of f. 8 The function g with domain 3 is defined b g() 6. Sketch the graph of g() and find the range of g. 9 The function f with domain 0 80 is defined b f() 2 sin. Sketch the graph of f() and find the range of f. 0 The function h with domain 0 is defined b h() cos 3. Sketch the graph of h() and find the range of h..5 Greatest possible domain A function is sometimes defined for all real values of. We sa the domain is the set of real numbers,, or. When this is the case, the domain is sometimes omitted and implicitl understood to be the set of real numbers. For instance f() (with no mention of a domain) implies that can take all real values. Sometimes restrictions on the domain are necessar. For instance f() 3 cannot be defined for 0. belongs to. If ou tr to find f(0) on our calculator ou will get an error message. Its greatest possible domain is therefore, 0. Since we cannot find square roots of negative quantities, the function g, where g() 3, does not eist for 3. The greatest possible domain for g is 3. EXERCISE D Determine the greatest possible domain for each of the following functions, f. f() 2 f() 3 ( )

9 C3: Functions 9 3 f() 3 4 f() 2 ( ) f() 6 f() 2 2 ( 4) 3 7 f() 8 f() ( )( 2) 4.6 Graphs that represent functions Consider the two graphs opposite. (a) In graph (a), for each value of ou can draw a vertical line and see that it gives a unique value. An horizontal line for a particular value of also corresponds to a unique value of. Graph (a) represents a one-one function. Repeating the procedure for graph (b). An vertical line gives one value of and so the graph represents a function. This time, however, some of the horizontal lines pass through more than one point on the curve and indicate that more than one value of maps onto a particular value of. (b) The function represented b graph (b) is man-one..7 Graphs that do not represent functions Contrast the graphs shown here with those in the previous section. For certain values of ou can draw a vertical line and see that it does not correspond to a unique value of. These graphs do not represent functions. For certain values of there is more than one value of. The corresponding mapping diagram would be one-man and cannot represent the function.

10 0 C3: Functions EXERCISE E For each of the following graphs, state whether it represents a function or not. For the functions, identif them as one-one or man-one Composite functions The term composition is used when one operation is performed after another operation. For instance: This function can be written as h() 5( 3). Sometimes ou have two given functions such as f and g and need to perform one function after another. Suppose f() 2 and g() 2 3,. What is f[g()]? g() 2 3 f[g()] f(2 3) (2 3) 2. The epression f[g()] is usuall written without the etra brackets as fg() and fg is said to be a composite function. 3 5( 3) Add 3 Multipl b 5 g() The function gf can be found in a similar wa. f() 2 so that gf() g[f()] = g( 2 ) 2 3 ( 2 ) The composite function gf is such that gf() g f fg() You could tr with a number g(2) f[g(2)] f(8) Although we write fg() the function g operates first on because it is closest to.

11 C3: Functions Worked eample.6 The functions f and g are defined for all real values of b f() 3 5 and g() 3 2. The composite functions fg and gf are such that p = fg and q = gf. Find p() and q(). Solution g() 3 2 f[g()] f[3 2] (3 2) 3 5 The composite function fg p. Hence p() (3 2) 3 5. f() 3 5 g(f()) g( 3 5) 3 2( 3 5) Since gf q, q() 3 2( 3 5) Note that fg is not the same as gf Although we could multipl out the brackets, it is best to leave the functions in this more compact form. EXERCISE F Assume that the domain of the functions in this eercise is the set of real numbers. Find an epression for fg() for each of these functions: (a) f() and g() 5 2, (b) f() 2 3 and g() 2, (c) f() 3 and g() 3, (d) f() 4 2 and g() ( ) 2. 2 For each pair of functions in question, find an epression for gf(). 3 Given that f() 2 2 and g() 3, find: (a) fg(), (b) gf(), simplifing our answers. 4 Given that f() 2 3, find (a) ff(2), (b) ff(a). Solve the equation ff(a) a. 5 Given that f() k 2 and g() 4 3, find in terms of k and : (a) fg(), (b) gf(). State the value of k for which fg() gf(). 6 Given that f() 2 and g() 5, find fg() and gf(). Show that there is a single value of for which fg() gf() and find this value of.

12 2 C3: Functions 7 The functions f and g are defined with their respective domains b 3 f(), 2 2 g() 2. (a) Find the range of g. (b) The domain of the composite function fg is. Find fg() and state the range of fg..9 Domains of composite functions In the eamples of composite functions considered so far the domains have been the real numbers, but sometimes the domains need more careful consideration. Consider the composite function fg g g() The whole of the range of g must be included in the domain of f, otherwise the domain of g needs restricting. f fg() Worked eample.7 Given the functions f and g such that f(), 0, and g() 5,, find the maimum possible domain of fg. Solution It is necessar to solve the inequalit g() 0 so that the range of g consists onl of positive numbers The maimum domain of fg is therefore 5. Initiall it might seem that an real number can be part of the domain of fg since the domain of g is. However g(7) 2, for eample, and this is not acceptable to be fed into f (see the diagram above)..0 Inverse functions The function f defined for all real values of b f() 3 4 can be thought of as a sequence of operations Multipl b 3 Subtract 4 If ou reverse the operations and the flow, Divide b 3 Add 4

13 C3: Functions 3 The new function, g sa, can be written as g() 4. 3 In general fg() f Also gf() g(3 4) (3 4) A function g such that fg() and gf() is said to be the inverse function of f and is denoted b f. In this case, f () 4. 3 Notice that f(3) 5 and g(5) 3. Similarl f( ) 7 and g( 7), etc. This is purel a smbol and should not be thought of as a reciprocal. A reverse flow diagram can be used to find an inverse function when occurs onl once in f(). You consider how f() has been constructed as a sequence of simple operations and set up a flow diagram. Then ou reverse each operation and reverse the direction of the flow to find f (). Worked eample.8 Find: (a) f () and (b) g (), where f() 3 5, and g(),, 2. 2 Solution (a) A flow diagram approach gives which when revised produces 3 5 So f () = 3. 5 Alternativel, f() 3 5 so let 3 5 Rearrange to make the subject of the equation Now interchange and 3 5 Hence f () Cube Add 5 5 Cube root Subtract The reverse flow diagram method can be used to find inverse functions in simple cases when occurs onl once in f(). This worked eample gives a more general method for finding inverse functions. The forward flow diagram has the boes cube then add 5. The reverse flow diagram would be subtract 5 then take the cube root giving the same answer for the inverse function.

14 4 C3: Functions (b) Since occurs more than once in the epression for g(), a flow diagram cannot be used. g() so let 2 2 ( 2) 2 2 ( ) 2 2 ( ) Now interchange and 2 ( ) Hence 2 g () ( ) You can check that the answer is correct b choosing a value of from the domain. For instance, when 3, g() gives g(3) = 3 = 4. Using the answer for g (), g (4) = 8 = = 3, which suggests the answer is correct. Multipl up b ( 2). Collect all the terms involving onto one side. Make the subject of the formula. Essentiall we are interchanging the domain and range to produce an inverse function. The domain for g is,. The inverse of f can be found b the following procedure: Write f(). Rearrange the equation to make the new subject. Interchange and (equivalent to reflecting in ). The new epression for is equal to f ().. Condition for an inverse function to eist In order for f to eist, the function f must be one-one. You can easil draw the graph of f when it eists. The graph of f is obtained from the graph of f() b reflection in the line provided ou have equal scales on the - and -aes. This is because the domain and range are interchanged when ou perform an inverse mapping. If a function were man-one, the inverse mapping would be one-man and, as ou have seen in Section.7, this could not be a function.

15 C3: Functions 5 Worked eample.9 The function f is defined b f() ( 2) 3, 0 and is sketched with equal scales on the aes. ( 2) 3 (a) Find the range of f. (b) State wh the inverse function f eists. 2 (c) Find f () and sketch the graph of f. (d) State the domain and range of f. Solution (a) f(0) 8 and the graph shows that the values of f() increase as increases. Range is f() 8. (b) For each value of f(), there is a unique value of. The function f is one-one. (c) f() ( 2) 3 so let ( 2) 3 Rearranging to make the new subject. 3 ( 2) 2 3 Interchanging and. 2 3 f () 2 3 Graph of f obtained b reflection of graph of f in line. (d) The domain of f is 8 (since the range of f is f() 8). The range of f is f () 0 (since the domain of f is 0)..2 Self-inverse functions Suppose ou take the reciprocal of 5. You get 0.2. Taking the 5 reciprocal of 0.2 gives 5. Doing the operation twice brings 0.2 ou back to the number ou started with. This is true for ever non-zero number ou tr to take the reciprocal of. Taking the reciprocal is an eample of a self-inverse operation. The function f defined for all non-zero values of b f() is called a self-inverse function. Consider the function g given b g() 2,. If the inverse of g is g, then g g () and g g(). But gg() g(2 ) 2 (2 ) 2 2. This proves that the inverse of g is itself g. Hence g is a self-inverse function. Alternativel g() 2 so let Interchange and. 2 g () 2,

16 6 C3: Functions ther eamples of self-inverse operations are divide into 6 and subtract from 7. Hence, when f() 6, ou can write f () 6 also. Similarl, when g() 7, then g () = 7 as well. You need to remember this if ou are using the flow diagram method to find the inverse function when some operations are self-inverse. Worked eample.0 The function f is defined for all non-zero values of b f() 3 2. Use a reverse flow diagram to find f (). Solution The function f can be thought of as a sequence of operations. If ou reverse the operations and the flow, 2 3 Divide into 2 Subtract from 3 3 Divide into 2 Subtract from 3 The inverse function f is given b f 2 () and is defined 3 for all real values of not equal to 3. EXERCISE G Each of the following functions, f, has domain. Find f () b means of a reverse flow diagram. (a) f() 5 7 (b) f() ( 2) 3 (c) f() (2 ) (d) f() (2 ) 3 6 (e) f() ( 2) 3 (f) f() For each of the functions in question, sketch the graphs of f() and f (). 3 The function f has domain 5 and is defined b 3 f(). 4 (a) Sketch the graph of f(). (b) Find the range of f. (c) The inverse of f is f. Find f ()

17 C3: Functions 7 4 The function f has domain 4 and is defined b f() ( 3) 2. (a) (i) Find the value of f(4) and sketch the graph of f(). (ii) Hence find the range of f. (b) Eplain wh the equation f() has no solution. (c) The inverse function of f is f. Find f (). [A] 5 For each of the following: (i) find the range of the function, (ii) find the inverse function, stating its domain, (iii) state the range of the inverse function. (a) f() (3 ) 3,, (b) g() 2,, (c) h() (2 3) 5, 0, (d) q() 5, 5, 4 (e) r(), For each of the following functions f and g: (i) find the range of the function, (ii) find the inverse function, stating its domain, (iii) state the range of the inverse function. (a) f() 2 5,, 3, 3 (b) g() 5 4,, (a) Sketch the graph of the function f, where f() 2 3,. Eplain wh f does not have an inverse function. (b) Sketch the graph of the function g given b g() 2 3,. Eplain wh g has an inverse and find g (). State the domain and range of g. 8 The function h has domain 0 and is defined b: h() 2 3 (a) Sketch the graph of h() and eplain wh h has an inverse. (b) Find h () and state the domain and range of h. 9 Determine whether an of the functions f, g and h are self-inverse functions. 2 (a) f(), 2, 2, (b) g() 3 4, 2, 2, (c) h() 3 5, 3, 3.

18 8 C3: Functions 0 The function f is defined b f() 2, 2. (a) Sketch the graph of f() and state the range of f. (b) Eplain wh the inverse function f eists and state its domain. Find an epression for f (). [A] The function f has domain 2 and is defined b f() 2 3. (a) Find f(2) and f(00). (b) Determine the range of f. (c) The inverse of f is f. Find f (). 2 The function f with domain 2 is defined b f(). 2 (a) Describe geometricall how the graph of, 0 is transformed into the graph of f(). (b) Sketch the graph of f(). (c) Eplain briefl wh f has an inverse function, state the domain of f, and epress f () in terms of. Worked eamination question The function f has domain 0 2 and is defined b f() 3. (a) Find f(0) and f(2). (b) Sketch the graph of f(). (c) Find the range of f. (d) State, with a reason, whether the inverse function, f, eists. (e) Find ff(), giving our answer in the form 9 a 6 b 3 c. Solution (a) f(0) 0 f(2) 8 9 (b) (2, 9) (0, ) The sketch is not intended to be an accurate plot and so ou should not get too worried about the relative positions of the two endpoints of the graph.

19 C3: Functions 9 (c) The lowest point on the curve is (0, ). The range, therefore, is f() 9. (d) Since the function is one-one, the inverse function does eist. (e) ff() f( 3 ) ( 3 ) Hence, ff() MIXED EXERCISE The function f has domain 2 and is defined b f() 2 5. (a) Find f( ) and f(2). (b) Sketch the graph of f(). (c) Find the range of f. (d) State, with a reason, whether the inverse function, f, eists. (e) Find ff(), giving our answer in the form 4 p 2 q. [A] 2 The functions f and g are defined with their respective domains b 6 f(),, 2 2 g() 2 2,. (a) Find the range of g. (b) The composite function fg is defined for all real values of. Find fg(), giving our answer in the simplest form. (c) The inverse of f is f. Find an epression for f (). (d) The graph of f() and the graph of f () intersect at two points. Find the coordinates of the two points. 3 The functions f and g are defined with their respective domains b 4 f(), 0 3 g() 9 2 2,. (a) Find fg(), giving our answer in its simplest form. (b) (i) Sketch the graph of g(). (ii) Find the range of g. (c) (i) Solve the equation g(). (ii) Eplain wh the function g does not have an inverse. (d) The inverse of f is f. (i) Find f (). (ii) Solve the equation f () f(). [A]

20 20 C3: Functions 4 The function f() with domain { : 0} is defined b 8 f(). 2 (a) Sketch the graph of f and state the range of f. (b) Find f (), where f denotes the inverse of f. (c) Calculate the value of for which f() f (). [A] 5 The functions f and g are defined b f() 3 4, g() 2,, 0. Write down, in a similar form: (a) the composite function fg, (b) the inverse function f. 6 (a) State which of the following graphs, G, G 2 or G 3, does not represent a function. Give a reason for our answer. [A] G G2 G 3 (b) The function f has domain 2 and is defined b f() 5. A sketch of f() is shown opposite. (i) Calculate f(2) and f(0). (ii) Find the range of f. (iii) The inverse of f is f. Find f (). [A] 7 The functions f and g are defined for all real values of b f() 5 3 g() 3 4. (a) Solve the inequalit f(). (b) The composite function fg is defined for all real values of. Find fg(), epressing our answer in the form p q 3, where the values of p and q are to be found. (c) The graph of g() is sketched opposite with equal scales on the - and -aes. Cop the graph of g() and, on the same aes, sketch the graph of g (). (d) Find an epression for g (). 2

21 C3: Functions 2 Ke point summar A function is a one-one or a man-one mapping. p3 2 The set of numbers for which a function is defined p3 is called the domain. 3 A function f consists of two things: p4 a defining rule such as f() 2 3; its domain. 4 The set of values the function takes for the given p4 domain is called the range. 5 When the domain of f is a continuous interval, the p5 range can be found b considering the graph of f(). The range consists of the possible values that can take. The range of f is written as an inequalit involving f(). 6 The composite function fg means first g then f, since: p0 fg() f[g()]. 7 A function f has an inverse onl when f is one-one. p2 Its graph is obtained b reflecting the graph of f in the line. 8 A reverse flow diagram can be used to find an inverse p3 function when occurs onl once in f(). You consider how f() has been constructed as a sequence of simple operations and set up a flow diagram. Then ou reverse each operation and reverse the direction of the flow to find f (). 9 The inverse of f can be found b the following p4 procedure: Write f(). Rearrange the equation to make the new subject. Interchange and (equivalent to reflecting in ). The new epression for is equal to f ().

22 22 C3: Functions Test ourself The function f is defined for all real values of b Section. f() Find: (a) f(0), (b) f(), (c) f( 2). 2 Find the range of the function f where f is defined b Section.3 f() 3, 3. 3 State the maimum possible domain for the function g where Section.5 g() 4. 4 The functions f and g are defined b f() 2 5 and Section.7 g() 7, and each has domain. Find an epression for gf() in its simplest form. 5 The function f with domain is defined b Section.0 f() (a) Find the range of f. (b) Find the inverse function, f, and state its domain. What to review 6 The function g is defined for all real values of, 4, b Section.0 g() Find the inverse function g and state its domain. Test ourself ANSWERS (a) 2; (b) 0; (c) 0. 2 f() gf() f () domain is all real values of, , (a) f() 9; (b) f () 3 2

### 5.2 Inverse Functions

78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

### Core Maths C3. Revision Notes

Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...

### Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

### Exponential and Logarithmic Functions

Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

### INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

### Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

### SAMPLE. Polynomial functions

Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

### Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

### 1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

### Higher. Polynomials and Quadratics 64

hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

### LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

### Graphing Linear Equations

6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

### Functions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study

Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic

.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

### D.3. Angles and Degree Measure. Review of Trigonometric Functions

APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

### DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

### I think that starting

. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

### When I was 3.1 POLYNOMIAL FUNCTIONS

146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

### 2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular

### Mathematical goals. Starting points. Materials required. Time needed

Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between

### More Equations and Inequalities

Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities

### 15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

### 7.3 Parabolas. 7.3 Parabolas 505

7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of

### Section 5-9 Inverse Trigonometric Functions

46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions

### 2.5 Library of Functions; Piecewise-defined Functions

SECTION.5 Librar of Functions; Piecewise-defined Functions 07.5 Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section.,

### STRAND: ALGEBRA Unit 3 Solving Equations

CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

### 10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

### In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)

Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and

### 2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle - in particular the basic equation representing

### Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

### 1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

1. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

### Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

### Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

### Five 5. Rational Expressions and Equations C H A P T E R

Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.

### 6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:

Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

### To Be or Not To Be a Linear Equation: That Is the Question

To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not

### 1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

### Core Maths C2. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

### Solving Absolute Value Equations and Inequalities Graphically

4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value

### Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

### 5.3 Graphing Cubic Functions

Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1

### SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

### Implicit Differentiation

Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some

### 2.3 TRANSFORMATIONS OF GRAPHS

78 Chapter Functions 7. Overtime Pa A carpenter earns \$0 per hour when he works 0 hours or fewer per week, and time-and-ahalf for the number of hours he works above 0. Let denote the number of hours he

### Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

### Why should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY

Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the -intercept. One real-world connection is to find the rate

### G. GRAPHING FUNCTIONS

G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression

### THE POWER RULES. Raising an Exponential Expression to a Power

8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar

### EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM

. Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,

### LIMITS AND CONTINUITY

LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

### The Distance Formula and the Circle

10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,

### Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

### Linear Equations in Two Variables

Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations

### Find the Relationship: An Exercise in Graphing Analysis

Find the Relationship: An Eercise in Graphing Analsis Computer 5 In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables.

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### y intercept Gradient Facts Lines that have the same gradient are PARALLEL

CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

### MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets

### MPE Review Section III: Logarithmic & Exponential Functions

MPE Review Section III: Logarithmic & Eponential Functions FUNCTIONS AND GRAPHS To specify a function y f (, one must give a collection of numbers D, called the domain of the function, and a procedure

### Linear Inequality in Two Variables

90 (7-) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.

### REVIEW OF ANALYTIC GEOMETRY

REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.

### LINEAR FUNCTIONS OF 2 VARIABLES

CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for

### Section 3-7. Marginal Analysis in Business and Economics. Marginal Cost, Revenue, and Profit. 202 Chapter 3 The Derivative

202 Chapter 3 The Derivative Section 3-7 Marginal Analysis in Business and Economics Marginal Cost, Revenue, and Profit Application Marginal Average Cost, Revenue, and Profit Marginal Cost, Revenue, and

### SECTION 7-4 Algebraic Vectors

7-4 lgebraic Vectors 531 SECTIN 7-4 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors

### 7.7 Solving Rational Equations

Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

### Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m

0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have

### Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x

OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial equations

### ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )

SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as

### Functions and their Graphs

Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers

### ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

### 3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?

Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using

### Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X

Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### Equation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1

Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows :- gradient = vertical horizontal horizontal A B vertical

### Trigonometry Review Workshop 1

Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions

### Connecting Transformational Geometry and Transformations of Functions

Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.

### 135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.

13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the

### Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

### So, using the new notation, P X,Y (0,1) =.08 This is the value which the joint probability function for X and Y takes when X=0 and Y=1.

Joint probabilit is the probabilit that the RVs & Y take values &. like the PDF of the two events, and. We will denote a joint probabilit function as P,Y (,) = P(= Y=) Marginal probabilit of is the probabilit

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### Florida Algebra I EOC Online Practice Test

Florida Algebra I EOC Online Practice Test Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end

### Ellington High School Principal

Mr. Neil Rinaldi Ellington High School Principal 7 MAPLE STREET ELLINGTON, CT 0609 Mr. Dan Uriano (860) 896- Fa (860) 896-66 Assistant Principal Mr. Peter Corbett Lead Teacher Mrs. Suzanne Markowski Guidance

### SECTION 5-1 Exponential Functions

354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational

### Slope-Intercept Form and Point-Slope Form

Slope-Intercept Form and Point-Slope Form In this section we will be discussing Slope-Intercept Form and the Point-Slope Form of a line. We will also discuss how to graph using the Slope-Intercept Form.

### You know from calculus that functions play a fundamental role in mathematics.

CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.

### The numerical values that you find are called the solutions of the equation.

Appendi F: Solving Equations The goal of solving equations When you are trying to solve an equation like: = 4, you are trying to determine all of the numerical values of that you could plug into that equation.

CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

### 1 Maximizing pro ts when marginal costs are increasing

BEE12 Basic Mathematical Economics Week 1, Lecture Tuesda 12.1. Pro t maimization 1 Maimizing pro ts when marginal costs are increasing We consider in this section a rm in a perfectl competitive market

### Core Maths C1. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

### 1 The Concept of a Mapping

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 1 The Concept of a Mapping The concept of a mapping (aka function) is important throughout mathematics. We have been dealing

### Integrating algebraic fractions

Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate

### LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,

LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are

### 5. Equations of Lines: slope intercept & point slope

5. Equations of Lines: slope intercept & point slope Slope of the line m rise run Slope-Intercept Form m + b m is slope; b is -intercept Point-Slope Form m( + or m( Slope of parallel lines m m (slopes

### SECTION 2-2 Straight Lines

- Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above

### Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals