Functions. 2.1 A mapping transforms one set of numbers into a different set of numbers. The

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1 Functions This chapter defines a function. It looks at was of combining functions to form composite functions as well as introducing the inverse of a function.. A mapping transforms one set of numbers into a different set of numbers. The mapping can be described in words or through an algebraic equation. It can also be represented b a Cartesian graph. Eample Draw mapping diagrams and graphs for the following operations: a add on the set {,,,, } b square on the set {,,,, } a Operation Mapping diagram Add Set A Set B Graph b Operation Mapping diagram Square Graph (set B) 0 8 (, 0) (, ) (, ) (, 7) (, 9) 0 8 (set A) (set B) (, ) (, ) (, ) This set is called the range. This set is called the domain. The range. The domain. 0 (set A)

2 CHAPTER Eercise A Draw mapping diagrams and graphs for the following operations: a subtract on the set {0,, 0,, } b double and add on the set {,,,, } c square and then subtract on the set {,, 0,,, } d the positive square root on the set {, 0,,, 9, }. Find the missing numbers a to h in the following mapping diagrams: 0 7 a c b 9 d e 7 g f h. A function is a special mapping such that ever element of set A (the domain) is mapped to eactl one element of set B (the range). A good wa to remember this is A B A B A B one-to-one function man-to-one function not a function You can write functions in two different was: f() OR f: This is the function double and add. Eample Given that the function g(), find: a the value of g() b the value of a such that g(a) c the range of the function.

3 Functions a g() () b g(a) a a a a c (0, ) Substitute in the formula. Substitute a and g(a). Sketch the function g(). The range is the values that takes. Range of g() is g() Eample Which of the following mappings represent functions? Give reasons for our answers. a b 9 O c d O O a b Ever element of set A gets mapped to two elements in set B. Therefore this is not a function. It is an operation (square root). Ever value of gets mapped to one value of. This is therefore a function. It can be written in the form a b for some constants a and b. This tpe of function is called

4 CHAPTER one-to-one because ever element of the domain goes to eactl one element in the range. c On the sketch of ou can see that 0 does not get mapped anwhere. Therefore not all elements of set A get mapped to elements in set B. Hence it is not a function. d This is a function. It is called a man-to-one function because two elements of the domain gets mapped to one element in the range. Eercise B Find: a f() where f() b g( ) where g() c h(0) where h : d j( ) where j : Calculate the value(s) of a, b, c and d given that: a p(a) where p() b q(b) 7 where q() c r(c) where r() ( ) d s(d) 0 where s() For the following functions i sketch the graph of the function ii state the range iii describe if the function is one-to-one or man-to-one. a m() b n() c p() sin() d q() State whether or not the following graphs represent functions. Give reasons for our answers and describe the tpe of function. a b c d e f

5 Functions. Man mappings can be made into functions b changing the domain. Consider O If the domain is all of the real numbers { }, then this is not a function because values of less than 0 do not get mapped anwhere. If ou restrict the domain to 0, all of set A gets mapped to eactl one element of set B. We can write this function f(), with domain {, 0}. Eample Find the range of the following functions: a f(), domain {,,, } b g(), domain {, } c h() domain {, 0 } State if the functions are one-to-one or man-to-one. a f(), {,,, } Here the domain is discrete as it onl has integer values. Draw a mapping diagram. 7 0 Range of f() is {,, 7, 0}. f() is one-to-one. b g() { } g() Here the domain is continuous. It takes all values between and. Sketch a graph. O Range of g() is 0 g(). g() is man-to-one.

6 CHAPTER c h() {, 0 } h() Sketch a graph. O Range of h() is h(). h() is one-to-one. Eample The function f() is defined b f() a Sketch f() stating its range. b Find the values of a such that f(a) 9. Note: This function consists of two parts one linear (for ), the other quadratic (for ). A useful tip in drawing the function is to sketch both parts separatel and also to find the value of both parts at. a f(), using f() f(), using f() 8 7 For, f() is linear. It has gradient and passes through on the ais. For, f() is a -shaped quadratic. You need to calculate the value of f() on both curves. 0 The range is the values that takes and therefore f(). Note that f() at so f() not f()

7 Functions b f() 9 For man questions on functions it becomes easier to sketch a graph. There are values such that f(a) 9. The positive point is where 9 The negative point is where The two values of a are and 7. Ignore as. Use both epressions as there are two distinct points. Eercise C The functions below are defined for the discrete domains. i ii Represent each function on a mapping diagram, writing down the elements in the range. State if the function is one-to-one or man-to-one. a f() for the domain {,,,, }. b g() for the domain {,, 9,,, }. c h() for the domain {,, 0,, }. d j() for the domain {,,,, }. The functions below are defined for continuous domains. i ii Represent each function on a graph. State the range of the function. iii State if the function is one-to-one or man-to-one. a m() for the domain { 0}. b n() for the domain { }. c p() sin for the domain {0 80}. d q() for the domain { }. 7

8 CHAPTER The following mappings f() and g() are defined b f() g() 9 9 Eplain wh f() is a function and g() is not. Sketch the function f() and find a f() b f(0) c the value(s) of a such that f(a) 90. The function s() is defined b s() a Sketch s(). b Find the value(s) of a such that s(a). c Find the values of the domain that get mapped to themselves in the range. The function g() is defined b g() c d where c and d are constants to be found. Given g() 0 and g(8) find the values of c and d. The function f() is defined b f() a b where a and b are constants to be found. Given that f() and f() 9, find the values of the constants a and b. 7 The function h() is defined b h() 0 { a}. Given that h() is a one-to-one function find the smallest possible value of the constant a. Hint: Complete the square for h().. You can combine two or more basic functions to make a new more comple function. g g() fg() f fg() means appl g first, followed b f. fg() is called a composite function. fg Eample Given f() and g(), find: a fg() b fg() c fg() 8

9 Functions a fg() f( ) b fg() f( ) c fg() f( ) ( ) g() f() g() f() g() f( ) ( ) Eample 7 The functions f() and g() are defined b f() and g(). Find: a the function fg() b the function gf() c the function f Note: f () () is ff(). d the values of b such that fg(b) a fg() f( ) g acts on first, mapping it to. ( ) b gf() g( ) ( ) 9 8 c f () f( ) ( ) 9 8 d fg() If fg(b) b b 8 b b f acts on the result. f trebles and then adds. Simplif answer. f acts on first, mapping it to. g acts on the result. g squares and then adds. Simplif answer. f maps to. f acts on the result. f trebles numbers then adds. Set up and solve an equation in b. 9

10 CHAPTER Eample 8 The functions m(), n() and p() are defined b m(), n(), p(). Find in terms of m, n and p the functions a b c a m() nm() b ( ) [n()] pn() c ( ) n( ) nn () mnn() mn () m() n() n() p() n() n() m() nn() n () Eercise D Given the functions f(), g() and h(), find epressions for the functions: a fg() b gf() c gh() d fh() e f () For the following functions f() and g(), find the composite functions fg() and gf(). In each case find a suitable domain and the corresponding range when a f(), g() b f(), g() c f(), g() If f() and g(), find the number(s) a such that fg(a) gf(a). Given that s() and t() find the number m such that ts(m). The functions l(), m(), n() and p() are defined b l(), m(), n() and p(). Find in terms of l, m, n and p the functions: a b c d e ( ) f g 7 0

11 Functions 7 8 If m() and n(), prove that mn(). If s() and t(), prove that st(). If f(), prove that f (). Hence find an epression for f ().. The inverse function performs the opposite operation to the function. It takes elements of the range and maps them back into elements of the domain. For this reason inverse functions onl eist for one-to-one functions. The inverse of f() is written f () f() f () ff () = f f() Function Inverse f() add subtract f () g() times divide b g () h() times, add subtract, divide b h () For man straightforward functions the inverse can be found using a flow chart. Eample 9 Find the inverse of the function h() 7. square square root Therefore, h () 7 Draw a flow chart for the function.

12 CHAPTER Eample 0 Find the inverse of the function f(), {, }, b changing the subject of the formula. Let f() (cross multipl) ( ) (remove bracket) (add ) (divide b ) Therefore f () f() f () f() Rearrange to make the subject of the formula. Define f () in terms of. Check to see that at least one element works. Tr. Note that f f(). f () Eample The function f() is defined b f() {, }. Find the function f () in a similar form stating its domain. f() f() The range of the function is the domain of the inverse function and vice versa. This fact can be used to solve man problems where the domain is not all of. O The range of the function is f() 0. Therefore the domain of the inverse function is 0. Sketch the function for the values of given. The range of the function the domain of the inverse function. Change the subject of the formula.

13 Functions The inverse function is f () {, 0} Alwas write our function in terms of. f () f() Note that the graph of f() is a reflection of f() in the line. This is because the reflection transforms to and to. 0 0 Eample If g() is defined as g() {, 0} a Calculate g (). b Sketch the graphs of both functions on the same set of aes. c What is the connection between the graphs? a b g () 0 g() g () Use a flow diagram. The domain of the inverse function is the same as the range of the function. (See graph below.) The range of the function is g(), so this is the domain of the inverse function. g() is a linear function with gradient passing through the ais at. Its domain is 0. g () is also a linear function with gradient 0. passing through the ais at. Its domain is.

14 CHAPTER c The graphs of g () and g() are mirror images of each other in the line. g() () g () 0 Eample The function f() is defined b f() {, 0}. a Find f () in similar terms. b Sketch f (). c Find values of such that f() f (). a b Let f() f () {, } 0 f() f () Change subject of formula. f() is a -shaped quadratic with minimum point (0, ). The range of the function is f(). The domain of the inverse function has the same values. First sketch f(). Then reflect f() in the line.

15 Functions c When f() f () f() 0 So This is easier than solving. b b ac Solve using a f() and f () meet where and are both positive. Eercise E For the following functions f(), sketch the graphs of f() and f () on the same set of aes. Determine also the equation of f (). a f() { } b f() { } c f() {, 0} d f() { } e f() {, 0} f f() { } Determine which of the functions in Question are self inverses. (That is to sa the function and its inverse are identical.) Eplain wh the function g() {, 0} is not identical to its inverse. For the following functions g(), sketch the graphs of g() and g () on the same set of aes. Determine the equation of g (), taking care with its domain. a g() {, } b g() {, 0} c g() {, } d g() {, 7} e g() {, } f g() 8 {, } The function m() is defined b m() 9 {, a} for some constant a. If m () eists, state the least value of a and hence determine the equation of m (). State its domain. Hint: Completing the square helps in these tpes of questions. 7 Determine t () if the function t() is defined b t() {, }. The function h() is defined b h() {, }. a What happens to the function as approaches? b Find h (). c Find h (), stating clearl its domain. d Find the elements of the domain that get mapped to themselves b the function.

16 CHAPTER 8 The function f() is defined b f() {, 0}. Determine a f () clearl stating its domain b the values of a for which f(a) f (a). Mied eercise F Categorise the following as i not a function ii a one-to-one function iii a man-to-one function. a b c O O O d e f O O O The following functions f(), g() and h() are defined b f() ( ) {, 0} g() { } h() { } a Find f(7), g() and h( ). b Find the range of f() and the range of g(). c Find g (). d Find the composite function fg(). e Solve gh(a). The function n() is defined b n() 0 0 a Find n( ) and n(). b Find the value(s) of a such that n(a) 0. The function g() is defined as g() 7 {, 0}. a Sketch g() and find the range. b Determine g (), stating its domain. c Sketch g () on the same aes as g(), stating the relationship between the two graphs.

17 Functions The functions f and g are defined b f: { } g: {, } Find in its simplest form: a the inverse function f b the composite function gf, stating its domain c the values of for which f() g(), giving our answers to decimal places. E The function f() is defined b f() a Sketch the graph of f() for. b Find the values of for which f(). E 7 The function f is defined b f: {, } a Find f (). b Find i the range of f () ii the domain of f (). E 8 The functions f and g are defined b f: {, } g: {, 0} a Find an epression for f (). b Write down the range of f (). c Calculate gf(.). d Use algebra to find the values of for which g() f(). E 9 The functions f and g are given b f: {, } g: {, 0} a Show that f(). ( ) ( ) b Find the range of f(). c Solve gf() 70. E 7

18 CHAPTER Summar of ke points A function is a special mapping such that ever element of the domain is mapped to eactl one element in the range. not a function man-to-one function one-to-one function A one-to-one function is a special function where ever element of the domain is mapped to one element in the range. Man mappings can be made into functions b changing the domain. For eample, the mapping positive square root can be changed into the function f() b having a domain of 0. If we combine two or more functions we can create a composite function. The function below is written fg() as g acts on first, then f acts on the result. For eample, Similarl g(), f() fg() f( ) f() fg() ( ) The inverse of a function f() is written f () and performs the opposite operation(s) to the function. To calculate the inverse function ou change the subject of the formula. For eample, the inverse function of g() is g () The range of the function is the domain of the inverse function and vice versa. 7 The graph of f () is a reflection of f() in the line. f() () f () a a 8