Higher. Functions and Graphs. Functions and Graphs 18

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1 hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms 5 7 Radians 6 8 Eact Values 6 9 Trigonometric Functions 7 0 Graph Transformations 7 HSN00 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For more details about the copright on these notes, please see

2 Unit Functions and Graphs UTCME Functions and Graphs Sets In order to stud functions and graphs, we use set theor. This requires some standard smbols and terms, which ou should become familiar with. A set is a collection of objects (usuall numbers). For eample, { 5,6,7,8} brackets). S = is a set (we just list the objects inside curl We refer to the objects in a set as its elements (or members), e.g. 7 is an element of S. We can write this smbolicall as 7 S. It is also clear that 4 is not an element of S; we can write 4 S. Given two sets A and B, we sa A is a subset of B if all elements of A are also elements of B. For eample, { 6,7,8 } is a subset of S. The empt set is the set with no elements. It is denoted b { } or. Standard Sets There are common sets of numbers which have their own smbols. Note that numbers can belong to more than one set. N natural numbers counting numbers, i.e. N = {,,, 4, 5, }. W whole numbers natural numbers including zero, i.e. W = { 0,,,, 4, }. Z integers positive and negative whole numbers, i.e. Z = {,,, 0,,, }. Q rational numbers can be written as a fraction of integers, e.g. 4,, 0 5,. R real numbers all points on the number line, e.g. 6,,,, 0 5. Page 8 HSN00

3 Unit Functions and Graphs Notice that N is a subset of W, which is a subset of Z, which is a subset of Q, which is a subset of R. These relationships between the standard sets are illustrated in the Venn diagram below. R Q Z W N EXAMPLE List all the numbers in the set P = { N :< < 5}. P contains natural numbers which are strictl greater than and strictl less than 5, so: P = {,, 4 }. Note In set notation, a colon ( : ) means such that. Functions A function relates a set of inputs to a set of outputs, with each input related to eactl one output. The set of inputs is called the domain and the resulting set of outputs is called the range. f f ( ) domain range A function is usuall denoted b a lower case letter (e.g. f or g ) and is defined using a formula of the form f ( ) =. This specifies what the output of the function is when is the input. For eample, if f ( ) = + then f squares the input and adds. Page 9 HSN00

4 Unit Functions and Graphs Restrictions on the Domain The domain is the set of all possible inputs to a function, so it must be possible to evaluate the function for an element of the domain. We are free to choose the domain, provided that the function is defined for all elements in it. If no domain is specified then we assume that it is as large as possible. Division b Zero It is impossible to divide b zero. So in functions involving fractions, the domain must eclude numbers which would give a denominator (bottom line) of zero. For eample, the function defined b: f ( ) = 5 cannot have 5 in its domain, since this would make the denominator equal to zero. R. This is read as all belonging to the real set such that does not equal five. The domain of f ma be epressed formall as { : 5} Even Roots Using real numbers, we cannot evaluate an even root (i.e. square root, fourth root etc.) of a negative number. So the domain of an function involving even roots must eclude numbers which would give a negative number under the root. For eample, the function defined b: f ( ) = 7 must have 7 0. Solving for gives 7, so the domain of f can be epressed formall as { : } EXAMPLE R A function g is defined b g ( ) =. + 4 Define a suitable domain for g. We cannot divide b zero, so 4 R.. The domain is { : 4} Page 0 HSN00

5 Unit Functions and Graphs Identifing the Range Recall that the range is the set of possible outputs. Some functions cannot produce certain values so these are not in the range. For eample: f ( ) = does not produce negative values, since an number squared is either positive or zero. Looking at the graph of a function makes identifing its range more straightforward. If we consider the graph of = f ( ) (shown to the left) it is clear that there are no negative -values. The range can be stated as f ( ) 0. Note that the range also depends on the choice of domain. For eample, if the domain of f ( ) = is chosen to be { R : } then the range can be stated as f ( ) 9. EXAMPLE = f ( ). A function f is defined b f ( ) = sin for R. Identif its range. Sketching the graph of = f ( ) shows that sin onl produces values from to inclusive. = sin This can be written as f ( ). Page HSN00

6 Unit Functions and Graphs Composite Functions Two functions can be composed to form a new composite function. For eample, if we have a squaring function and a halving function, we can compose them to form a new function. We take the output from one and use it as the input for the other. The order is important, as we get a different result in this case: Using function notation we have, sa, f ( ) = and g ( ) =. The diagrams above show the composite functions: g ( f ( )) = g ( ) f ( g ( )) = f ( ) = ( ) = =. 4 EXAMPLES. Functions f and g are defined b f ( ) = and g ( ) =. Find: (a) f ( ) (b) f g ( ) (c) g f ( ) (a) f ( ) = ( ) = 4. ( ) ( ) = ( ) (b) f g ( ) f = ( ). ( ) ( ) = ( ) (c) g f ( ) g =.. Functions f and g are defined on suitable domains b f ( ) = + and g ( ) =. ( ) ( ) ( ) ( ) Find formulae for h ( ) = f g ( ) and k ( ) = g f ( ). ( ) h ( ) = f g ( ) = f ( ) ( ) = +. square halve halve square k ( ) = g f ( ) = g + =. + 4 Page HSN00

7 Unit Functions and Graphs 4 Inverse Functions The idea of an inverse function is to reverse the effect of the original function. It is the opposite function. You should alread be familiar with this idea for eample, doubling a number can be reversed b halving the result. That is, multipling b two and dividing b two are inverse functions. The inverse of the function f is usuall denoted f (read as f inverse ). ( ) ( ( )) The functions f and g are said to be inverses if f g ( ) = g f =. This means that when a number is worked through a function f then its inverse f, the result is the same as the input. f f ( ) For eample, f ( ) = 4 and g ( ) + f ( g ( )) = f 4 + = 4 4 = + = f + = are inverse functions since: 4 g ( f ( )) = g ( 4 ) ( 4 ) + = 4 4 = 4 =. Page HSN00

8 Unit Functions and Graphs Graphs of Inverses If we have the graph of a function, then we can find the graph of its inverse b reflecting in the line =. For eample, the diagrams below show the graphs of two functions and their inverses. = = = f ( ) = g ( ) = f ( ) = g ( ) 5 Eponential Functions A function of the form f ( ) = a where a, R and a > 0 is known as an eponential function to the base a. We refer to as the power, inde or eponent. 0 Notice that when = 0, f ( ) = a =. Also when =, f ( ) = a = a. Hence the graph of an eponential alwas passes through ( 0, ) and (,a ) : = a, a > = a, 0 < a < (, a) (, a) EXAMPLE Sketch the curve with equation = 6. The curve passes through ( 0, ) and (, 6 ). = 6 (, 6) Page 4 HSN00

9 Unit Functions and Graphs 6 Introduction to Logarithms Until now, we have onl been able to solve problems involving eponentials when we know the inde, and have to find the base. For eample, we can 6 solve p = 5 b taking sith roots to get p = 6 5. But what if we know the base and have to find the inde? To solve 6 q = 5 for q, we need to find the power of 6 which gives 5. To save writing this each time, we use the notation q = log65, read as log to the base 6 of 5. In general: log a is the power of a which gives. The properties of logarithms will be covered in Unit utcome. Logarithmic Functions A logarithmic function is one in the form f ( ) = log a where a, > 0. Logarithmic functions are inverses of eponentials, so to find the graph of = log a, we can reflect the graph of = a in the line =. = log a ( a,) The graph of a logarithmic function alwas passes through (, 0 ) and ( a,). EXAMPLE Sketch the curve with equation = log6. The curve passes through (, 0 ) and ( 6, ). = log 6 ( 6,) Page 5 HSN00

10 Unit Functions and Graphs 7 Radians Degrees are not the onl units used to measure angles. The radian (RAD on the calculator) is an alternative measurement which is more useful in mathematics. Degrees and radians bear the relationship: π radians = 80. The other equivalences that ou should become familiar with are: 0 = π 6 radians 45 = π radians 60 = π 4 radians 90 = π radians 5 = π 4 radians 60 = π radians. Converting between degrees and radians is straightforward. To convert from degrees to radians, multipl b π and divide b 80. To convert from radians to degrees, multipl b 80 and divide b π. For eample, 50 = 50 π = 5π radians Degrees 80 π 80 π Radians 8 Eact Values The following eact values must be known. You can do this b either memorising the two triangles involved, or memorising the table. DEG RAD sin cos tan π 6 45 π 4 60 π 90 π 0 Tip You ll probabl find it easier to remember the triangles. Page 6 HSN00

11 Unit Functions and Graphs 9 Trigonometric Functions A function which has a repeating pattern in its graph is called periodic. The length of the smallest repeating pattern in the -direction is called the period. If the repeating pattern has a minimum and maimum value, then half of the difference between these values is called the amplitude. period ma. value amplitude min. value The three basic trigonometric functions (sine, cosine, and tangent) are periodic, and have graphs as shown below. = sin = cos = tan Period = 60 = π radians Amplitude = Period = 60 = π radians Amplitude = Period = 80 =π radians Amplitude is undefined 0 Graph Transformations The graphs below represent two functions. ne is a cubic and the other is a sine wave, focusing on the region between 0 and 60. ( p, q) = g ( ) In the following pages we will see the effects of three different transformations on these graphs: translation, reflection and scaling = sin Page 7 HSN00

12 Unit Functions and Graphs Translation A translation moves ever point on a graph a fied distance in the same direction. The shape of the graph does not change. Translation parallel to the -ais f ( ) + a moves the graph of f ( ) up or down. The graph is moved up if a is positive, and down if a is negative. (,) a is positive = g ( ) + ( p, q + ) a is negative = g ( ) Translation parallel to the -ais f ( + a) moves the graph of f ( ) left or right. The graph is moved left if a is positive, and right if a is negative. a is positive a is negative = g ( + ) = g ( ) ( p, q) (,) = sin + (, ) ( p, q ) ( p +, q) (, ) = sin 5 = sin( + 90 ) = sin( 90 ) Page 8 HSN00

13 Unit Functions and Graphs Reflection A reflection flips the graph about one of the aes. When reflecting, the graph is flipped about one of the aes. It is important to appl this transformation before an translation. Reflection in the -ais f ( ) reflects the graph of f ( ) in the -ais. = g ( ) ( p, q) = sin Reflection in the -ais f ( ) reflects the graph of f ( ) in the -ais. ( p, q) = g ( ) = sin( ) From the graphs, sin( ) = sin Page 9 HSN00

14 Unit Functions and Graphs Scaling A scaling stretches or compresses the graph along one of the aes. Scaling verticall kf ( ) scales the graph of f ( ) in the vertical direction. The -coordinate of each point on the graph is multiplied b k, roots are unaffected. These eamples consider positive k. k > stretches 0 < k < compresses ( p, q) = g ( ) ( p, q) = g ( ) = sin = sin Negative k causes the same scaling, but the graph must then be reflected in the -ais: = g ( ) ( p, q) Page 0 HSN00

15 Unit Functions and Graphs Scaling horizontall f ( k ) scales the graph of f ( ) in the horizontal direction. The coordinates of the -ais intercept sta the same. The eamples below consider positive k. k > 0 < k < = g ( ) = g ( p, q ) ( p, q) ( ) 6 = sin = sin Negative k causes the same scaling, but the graph must then be reflected in the -ais: = g ( ) p, q ( ) Page HSN00

16 Unit Functions and Graphs EXAMPLES. The graph of = f ( ) is shown below. = f ( ) 0 (, 4) Sketch the graph of = f ( ). 5 Reflect in the -ais, then shift down b : = f ( ) ( 5, ) 0 (, 6). Sketch the graph of = 5cos( ) where = 5cos( ) Remember The graph of = cos : 60 Page HSN00

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