Normally relation deals with matching of elements from the first set called DOMAIN with the element ofthe second set called RANGE.

Definitions

AA function is a relation with a property that for each element in the domain there is only one corresponding element in the range or co- demain Therefore funetions are relations but not all relations are functions Representation of a Function ‘The Concept of a Functions Pictorially Explain the concept ofa functions pietorially

Example 1

Which of the following relation are functions? [ \ Ls \ f _—=——

• Lic

\ ——__., \ sgh J

Solution

4, Itismota fmetion since 3 and 6 remain unmapped. Functions

TESTING FOR FUNCTIONS,

Beale qa) y s 1s a function. isa function @ y

• x

itis nota function ~ I 7 x [ itis a function

• ii

\K } x an : +x

• Which of the following relations are functions?

(a) (2 e" e \ &]} \ 5 (bo) / = = e \ {ys \ jt

a fOxypx<y} b. LOx.y)ix> yl e fOny) y=x} Domain and Range of'a Function The Domain of a Function Ft State the domain ofa function ry =f (x)shat i y is function of x then domain is a set of x values that satisfy the equation y

‘The Range of Function State the range of function Lyf (0) that is y is a function of x , then therange isa set of y value satisfying the equation y ~ FO).

Example3

Solution

Therefore y is found in between ~ 11 and 4

Example 4

Solutions

Domain = all real numbers Range: fixyeyax?-3 Make x the subject yeaa? Wwts=x Since there is no square root of negative number(s) then y + 3 must be greater than or ‘equal to zor0 ie y +320 ye-3

• Range= {y:y2-3 }
• For each of the following functions, state the domain and range

• for cach of the following funetions state the domain and range

a fayex bfx ats2 © nett 4 rie The range ofthe function fix) 3—2x for 0°0 x £7is: ay -l8ey 83 by Beyels © y3fyels ad yee yea

• The range of the function

fix)-2e+1 is y:-3E y £17 what is the domain of this function’ aoe 3ExE17 bx DEES © xcITEXB

• Which of the following relations represents a function:

a Re(x yey fore 20

• Which of the following relations is a function:

a £62)<F)

b PEP FEA)

The function crosses , y— axis at Evy ei tit mny i fica One to functions; Aone to one function is a function in which one element from the domain i mapped to exacily ami element inthe range That is if a 4b then f (a) #f(b) \ atl Vs] \ °F waa A (pe to ne foneson Many to one fanction; Ths is ahather type of faction wifi propery iat tw or nore cements fiom the domi ca have one image (the same image), ie {@)=F(b) butash i~ aN

/ a—\— ie" \ el pal \ J \ /

Examples of one to one fictions

2 fQx)axt6

• f(x) =x" + Lete

aes oi t ob Rani 1 fix) 1

reaps

ie f(@)=f(b) but ash { /… \ [ =\— + rn

• Fy uf

\ J ee] \ \ J Lemp

Solution

Van sa o™ / / Y / .———_- 3 \ 3 _j A, vl \ / \ / NB: In example 1, ffx) is not a one to one function because -2 and 2 in A have the same image in B, thats 4 isthe image of both 2and.2.

Also 1 isthe image of both 1 nd -1 ape State whether or not if the following graphs represent a one to one function:

• ¥ () y

\ w 78) \ x

Solution

then, the Function is many to one and if it erases at only one point then the graph represents a (a) y _\ f a Sy, lf 7 x [ (x) is many to one. v

• /7* We)

.b{x) is one to one.

Graphic Function

Graphs of Functions

Draw graphs of finetions Many functions are given as equations, this being the case, drawing a graph of the equation is obtaining the graph of the equation which defines the function Note that, you can draw a graph of a function if you know the limits of its independent variables as well as dependent variables. i.e you must know the domain and range of the given function

Example 8

Draw the graph of the following functions

b. g(sjax7-2e-1 © hayes

Solution

‘The domain and range of Fare the sets ofall real numbers

Table of value PP " Pet w f 3} [foxy Lo yy eee a a fs, fl [+r fot

ye 21 ere rane

Yaintercept is -1

Other value are tabulated as follows a Graph: y \ 5 jo) SY 41/414 1___,x 643 42 AS 1A 304 °5 a forh(x) = x*

Solution

Table of Values

x 2 4 o 1 2 roe ee Graph 12 fu 4 J ttt tt ae ee ae 654327) 123 4 567 i 4 The first graph is the graph of linear function, the second one is called the graph of a quadratie function and the last graph is for eubic Funetion

Example9

Draw a graph of the function:

Solution

a am}, b6, oT

Other values x ae ee . Fats} ef a q vax J ™ a / \ +++ + 4 + H+ ttt tx 6543 24 123 456 7 i {s lg LWhich o°the following are ane to one function? a Raya 3x—x? b gaan ce ktsyax A fixjexttn’ 2 kerx!

4s

• Draw the graph of the following fmetions

b h@extt © gojaxtxts

• At what values of x does the graph ofthe funetion x) = x26 cross thex- axis?

a xeSand x7 b. xBand 6 © xvSandx-2 d xdand x1 4, Which ofthe following function is one to one function? a x2 b fayaxta? a7 de fixpen*+xt2 Funetions with more than one part Some funetions consist of more than one part. When drawing their graphs draw the pars separately.

the graph includes an end point, indicate it with a solid dot iit does not include the end point indicate i witha hollow dot. E.g. draw the graphs ofthe functions (@) Fla) x#1 for 0

• x)= Hor x0

(a) G rs Ae +44 x 2 Aaa 1 2 (b) y a Pa a ie

Example 10

Solved a

• Let fox) i if x<0)}

40 f x0 } , it wo ) (@)Find f(-8)

• Find £ (16)
• Sketch its graph

(@) State the domain and range of f

Solution

• £(-8 means f(x=8)

Bul-8<0 and if is stated that when x<0

Graph. Y

q = tox 3 2A 123

Sketch the graph ofeach of the following functions and foreach ease sate the domain and range

• Fo) j if xq

0 if { 2 wd (FO) [oxetit xed 2 if x+1<0)

<1 for O-Osxs1 4 for «2 Absolute value functions (Modulus functions) The absolute function 1s defined

So if x is positive, x remains unchanged and if x is negative, x changes to —« which also becomes positive. Hence x is always grater or equal to Zero, Lo. x 20 for all real values of x.

We can obtain the graph of f(x) = x as follows; so Table of values a ec La La

oP Pe PP

Graph y \ q Vis \ J / ‘a / 4324 2 3 4 A x)

Example 11

Solve the following

{a) sketch the graph of f(x)

• State its range

si

Solution

table of values. [-r fF FP PP Pf é 6 wo 5 Abi x al S/o \ 4 So 1s x 43 24:1 23 4

• Range = {yeve 0 }

When sketching a modulus function itis helpful to find first the minimum or maximum value of y. ‘Suppose the function is given as, y= aly — ble

The value of y becomes,

So (x, y) = (b, c) which is the vertex of the function Step functions For any number x, [x] is the value of x when rounded down to the integer below or equal to it So [x] is the greatest intogor which is less than or equal to x.

When drawing the graph of y = [x] the graph is horizontal between integers.

Example 12

Draw the graph of

Solution

sa g i oa) 4 3 2 A 1 2093 4 5.6 oad —o 3

• FO) =e + 2143 (b) x) =2-121

• Given that f(x) = [xl

Find (a) f (-6.5) (0) f (12.01)

Find (a) f 2.6) (b) (3.3)

• Draw the graph of the function f(x) = [x] + 3 and state its domain and range
• Sketch the graph of the functions (a) y = [2x] (b) y= [x-3]

Inverse of a Function The Inverse of a Function Explain the inverse of a function In the diseussion about relation we defined the inverse of relation This rue that the inverse of the relation is also a relation Similarly because a function is also relation then every function has its inverse The Inverse of a Function Pictorially Show the inverse of a function pictorially According to the definition of function the inverse of a function is also a funetion if and only if

the function is one to one / \ i% a \ The inverse is also function if \ J ® 7\ vA \ The inverse is not a function, / Ina \ pe a oa . _ A \ 7 \ J The inverse of a function given in (a) is shown below, ™* rr.

if ay en G / Its inverse is also a function And the inverse of the function given in (b) is { \ f™ [ ==———— /\ $f. The inverse is NOT a function bee \ J te The Inverse of a Function ser Le. Ify=fx), then x =f" (y)

Example 13

2 ROY

Solution

Fe

• f(x) =3x-8

mS

() fx) =y=? yex Inverse: x= y® Then y=Vx £00 Vx A Graph of the Inverse of a Function Draw a graph of the inverse of a function

Example 14

domain and range of F(x)

solution

so

yx

yoKe5

Graph of 1 (x): Tabio of values peje Y 5 d ff 5 OA 7 fo 6 ff. ff 2 fo…" x $4 3 24 123 4 a Domain = {All real numbers} Range = {All real numbers} NB: ifa function f takes a domain A to a range B, then the inverse f takes B back to A.

Hence the domain of fis the range off, and the range off" is the domain off. ‘The Domain and Range of Inverse of Functions State the domain and range of inverse of functions

Example 15

Solve

Where x is real number. Find the domain and range of f

Solutions

The function takes all values of y between

The domain of fis the range of f and the range of fis the domain of f*

1LFind the inverse of exch ofthe following functions: ot

• FQ) =3×2+8
• FQ) =x-9
• given that f(x) =7x-4 find (8)
• given that f(x) = 3x+4 for 3x <8

Find f(x) and state the domain and range of f"(x)

• Lot f(x) =x°-1

Evaluate (1) F(6) (i) P(e)

• given that f(x) = x2-2"! 43, what is the value of f (-4)?
• 15 (b) 11 (p27 5 { )

(a)4 ()6 (C8 (a { )

• given that f(x) = x — 5 -7, what is the minimum value of fo)?

(a)7 (0)0 €)5 7 ( )

• Let 1(x) =2x -¥3 what Is the value of x such that #-1 (x) =2?

@? > @ ( )

a

• (€) f00= 2 (%+3) (A) 100)=543

a