Modifying Functions - Families of Graphs

Transcription

1 Worksheet 47 Modifing Functions - Families of Graphs Section Domain, range and functions We first met functions in Sections and We will now look at functions in more depth and discuss their domain and range more formall Defining Functions A function f is specified b a rule and two sets These sets are the domain, previousl discussed in worksheet, which we ll call A, and the codomain which we ll call B The domain of a function contains all the values that we can input into the function The codomain is a set containing all possible values the function could achieve The codomain is usuall given We can think of our function as a mapping from the domain to the codomain and we usuall write this as f : A B where A = domain B = codomain Eample : Consider f : R R, where f() = We sa that f takes real numbers (from the domain), applies the rule and maps them to other real numbers in the codomain Here the rule sas that we square what we put into our function So for eample f() = 4 That is, f maps from the domain to 4 in the codomain Another eample is f( ) = In this case f maps in the domain to in the codomain Eample : Consider f : [, ) [, 0), where f() = + The above function f has domain [, ) and codomain [, 0) The rule of this function is that we take some element of the domain and then evaluate + to find f() For eample f( ) = 4 The last set we will need to define is the range of a function The range, also talked about in worksheet, is a subset of the codomain, and contains all of the values that f actuall attains The range is alwas contained in the codomain, but the codomain might not necessaril be in the range (it can contain more things) In Eample the range of f is the same as the codomain, [, 0)

2 Eample : Consider again f : R R where f() = Since we square all our inputs, we will never get a negative number as an output In other words, f() will onl attain 0 and positive values So the range of f() is [0, ) Notice this is a subset of the given codomain R Eample 4 : Consider f : [0, ) R, where f() = 7 Here the domain is [0, ), the range is [ 7, 0) and the codomain is R (given) Note that the range is a subset of the codomain All functions have onl one value of f() for each value of Eample 5 : Find the domain of f() = 5 We can never divide b 0, thus for to be in the domain it must satisf 5 0 So the domain of the function is 5 Set Notation Another wa of writing the domain from Eample 5 is to use set notation We sa Domain = { R : 5 }, which we read as the set of all real numbers, such that is not equal to 5 We use the brackets { } to respresent our set and the smbol to mean belongs to The colon : ma be read as either such that or with the propert that Eample 6 : Find the domain and range of f() = + We can onl take the square root of positive numbers or 0, so to be in the domain must satisf + 0, which gives domain [, ) The square root function alwas gives a positive or 0 result, so + 0 so the range of the function is [0, ) We can write this using set notation as Domain = { R : } Range = { R : 0 }

3 Eercises: State the domain and codomain and find the range of the following functions (a) f : (0, 5) R, f() = (b) g : [, 7] R, g() = + Find a suitable domain and the related codomain for f() = Find the domain of the functions below and epress it in set notation (a) f() = (b) f() = (c) f() = (d) f() = (e) f() = 7 (f) f() = 6 4 Find the domain and range of (a) f() = + (b) f() = + 9 (c) f() = 4 Section Modifing Functions Translations You ma have noticed that it is often easier to find the range of a function b looking at its graph In fact the graph of a function can give us the big picture on how the function behaves The net two sections look at modifing known functions to quickl and easil sketch other graphs We know how to draw the standard graphs of some basic functions, for eample

4 = = = f() = f() = This worksheet will show ou how to easil and quickl draw modified versions of these graphs The first kind of modification is one that occurs on the -coordinate For eample we might want to sketch = This graph is taken b picking some value of, squaring it and then subtracting This is the same for ever value of, so this is just the graph = shifted down b units = = = 4

5 Definition : The modification = f() + a is drawn b shifting the graph of = f() up b a units Similarl the modification = f() b is drawn b shifting the graph down b units Eample : Sketch the graphs = + and = The first graph is a shift up of units, and the second graph is a shift down of one unit = + f() = The other kind of modification is one that occurs on the coordinate For eample suppose we want to sketch (a) = ( ) (b) = + (c) = The ke thing to notice here is that the order of operations tells us that in (a) we subtract from and then square the epression Similarl in (b) we add to and then appl the function Finall in (c) we subtract from and then take the square root All of these amount to essentiall the same thing, ie we are adding or subtracting a number before appling the basic function To see this, think for a moment about how the graphs are drawn Let s look at (a) - going from the original function = to the new function = ( ) In the original function we take some particular value for, let s sa = 4, and then square it to get the value So we have a point on the original graph (4, 6) In comparison, looking at our new function and taking the same value, we subtract from it before squaring so the value is = = 9 and the point on our new graph is (4, 9) So the value is the same as the value unit to the left in the non-modified graph 5

6 = ( ) = ( + ) A simple wa to state a rule would be to sa: Definition : For a modification to a function on the coordinate of f(+a) the graph is drawn b shifting f() to the left b a units Similarl the graph f( b) is drawn b shifting f() to the right b b units The other graphs look like: - f() = + f() = The modifications we have looked at in this section have either shifted our original function verticall or horizontall We call this kind of modification a translation Eercises: On the same diagram sketch the graphs of = and = + Sketch (a) = + 5 (b) = + 5 (c) = (d) = + 0 (e) = 7 (f) = + 7

7 Section Other Modifications There are three more standard modifications to consider, the first is multipling the function b a constant This modification takes the original values of the functions and changes them b the scalar that we are multipling b Eample : = ( ) In this eample the number is the scalar of multiplication = ( ) The onl difference between this eample and the previous drawing of = ( ) is that this function is steeper The net modification we ll talk about is taking absolute values This modification takes an positive values and leaves them unchanged, and takes an negative values making them positive with the same value This is the same as putting a mirror along the ais, and drawing an values below the ais at their mirrored position above the ais Eample : Sketch the graph of = = = 7

8 The last modification we will look at is taking reciprocals The reciprocal of is, we just flip the fractions over (thinking of as ) Here are some numbers and their reciprocals Number () Reciprocal Number () Reciprocal Note that the smaller the number, the larger the reciprocal and the larger the number the smaller the reciprocal Note that 0 has no reciprocal, but as gets closer to 0 its reciprocal gets larger (approaches ) If it gets close to zero and is negative then its reciprocal will approach, if it is positive then it approaches Similarl as gets large (approaches ) the reciprocal approaches 0 Eample : We can see how to draw the reciprocal function = graph = = f() = b using the On the graph of = notice there is a break along the line = 0 This is because cannot take the value of 0, however as gets closer to 0, approaches ± respectivel We call the line = 0 a vertical asmptote 8

9 Eample 4 : Sketch the graph of = = = Eercises: Sketch (a) = ( ) + (b) = (c) = (d) = Sketch the graph of = Using this and considering reciprocals sketch the graph of = Let f() = Sketch (a) = f() + (b) = f() (c) = f( + ) (d) = f() 4 Sketch f() = + and find its domain and range 9

10 Eercises for Worksheet 47 Sketch the following functions and find their domains and ranges (a) = (b) = 7 (c) = 4 (d) = + 6 (Hint: complete the square) Sketch the following functions (a) = + (b) = ( + ) (c) = + ( + ) (d) = ( + ) (e) = (+) (f) = 5( + ) Sketch the following functions (a) = (b) = 4 (c) = + (d) = 4 4 In this question f() = Sketch the following (a) = f() (b) = f( + ) (c) = f() + (d) = f() + (e) = f() 0

11 Answers for Worksheet 47 Section (a) Domain is (0, 5), range is (, ), codomain is R (b) Domain is [, 7], range is [0, 6], codomain is R A suitable domain is [, ) and a suitable codomain is [0, ) (a) { R : 0 } (b) { R : > 0 } (d) R (c) { (e) { R : 7 } } R : (f) { R : 4 or 4 } 4 (a) Domain is R Range is [, ) (b) Domain is [ 9, ) Range is [0, ) (c) Domain is R Range is R