FUNCTIONS

Definition: If there is an association or a correspondence between each element x of a set X to exactly one

element y of a set Y then the association if a function from X to Y.

Alternative definition: A function, f, from X to Y is a set of ordered pairs of real numbers, yx, , in which not

two distinct ordered pairs have the same first number.

Notation: A function from a set X to a set Y is usually denoted by YXf :

REMARKS:

• The set X, the first set, is called the domain.

• The set Y, the second set, is called the co-domain.

• If an element y of a set Y corresponds to the element x of a set X, we call y the value of the function at

x or the image of x under the function, and we write is as follows: yxf read as “ yxf equals of ”.

• The set of values Yy which actually corresponds to some x in the domain is called the range of the

function f.

Illustration of a function:

Note:

• An element of the range may correspond to more than one element of the domain.

• Some elements in the set Y (co-domain) may not be in the range.

• An element x of the domain NEVER corresponds to more than one element in the co-domain.

The following are not illustrations of a function:

f: XY

X Y

f: GH

G H

f: FA

F A

f: XY

X Y

f: GH

G H

Determine if the following are functions or not. ( Ryx , ).

1. )5,1(),4,1(),3,1(),12(, NOT A FUNCTION

2. Ryxxyyx ,,32, FUNCTION

f: XY

X Y

1

2

3

4

5

Every point on this line satisfies the

equation 32 xy and hence are

elements of the given set. Note that

for every x there is only one

corresponding y value.

3. Ryxyxyx ,,932 ,22

Note that the equation 93222 yx represents a circle. Thus

it does not represent a function. See the illustration below:

Clearly if 1x , 223y , and thus violating the definition of a function.

Definition: The graph of a function f is the set of all points 2),( Ryx for which ),( yx is an ordered pain if

the function f .

One problem that we may ask when we are dealing with function is the question of the domain of definition.

What would be the largest possible domain that the given function may have?

In our study, we only limit ourselves with real valued functions. This means that we only consider domains

which are subsets of the real numbers. Now, it is difficult to determine the domain by trying all possible subsets

of R (the set of real numbers). So what we do is determine which number/s can be excluded from R and then

those which are not excluded are part of the domain. Consider the following examples.

Find the largest possible domain of the following functions and try to determine its range.

1. 4)( xxf . This would mean the xy . So if we consider the set of real numbers as our domain, every

value of x would give a real number as a value of the function. Thus the domain is the set of real numbers, that

is, domain RxxD | . Since xy , then the values of y are exactly those values of x, hence, the range is

also the set of real numbers RyyR | . We can also determine the domain and range of a function by

looking at its graph.

Note that the line extends indefinitely upwards and downwards. This would imply that the values of y cover all

real numbers.

2. xxf )( RxxxD ,0| or 0 RD

RyyyR ,0| or 0 RR

3. 2)( xxf Domain is the set of all real numbers.

RyyyR ,0| or 0 RR

Line 4)( xyxf

4. xxf )( Domain is the set of real numbers

RyyyR ,0|

5. 1)( xxf Domain is the set of real numbers

RyyyR ,0|

6. 1)( xxf Domain is the set of real numbers

RyyyR ,1|

xxf )(

1)( xxf

1)( xxf

7. 24)( xxf 22| xRxD

0| yRyR

Exercises:

Determine the largest possible domain for the following functions.

1. 52)( xxf 2. x

xxf

1)(

3. 3

52)(

x

xxf 4. 1)( 2 xxf

5. 5)( xxf 6. 3)( xxf

7. 1)( xxf 8. 4)( 2 xxf

Solutions:

1. D: all real numbers

2. 0\RD all real numbers except 0, since 0x would make the function undefined

3. 3\RD all real numbers except 3, since 3x would make the function undefined

4. D: all real numbers

5. D: all real numbers

6. D: all real numbers

7. 1| xxD . Since we want a real valued function, the expression inside the radical sign should be greater

than zero; hence, x should be greater than 1.

8. Since we want a real valued function, the expression inside the radical sign should be greater than zero;

hence, solving the inequality 042 x will give us, 2x or 2x , that is ),2[]2,( .

Function values.

Exercises:

Given the following functions, 52)( xxf and 1)( 2 xxg , determine the values of the function at the

specified value of x.

1. 2x , 154522)2( f , thus 1)2( f

3141)2()2( 2 g , thus 3)2( g

2. 0x , 550502)0( f , thus 5)0( f

1101)0()0( 2 g , thus 1)0( g

3. kx , 5252)( kkkf , thus 52)( kkf

11)()( 22 kkkg , thus 1)( 2 kkg

4. 1 dx , 32512)1( dddf , thus 32)1( ddf

ddddddg 21121)1()1( 222 , thus dddg 2)1( 2

Given the above functions, determine the following

1. 1f 2. )1(g 3. )10(f 4. )5(g

OPERATIONS ON FUNCTIONS

Examples:

Given the functions 2)( xxf , 2)( xxg and2

2)(

x

xxh , determine the following

1. ))(( xgf )()( xgxf

x

xx

2

22

2. )()()( xgxhxgh 22

2

x

x

x

Definition: If f and g are function, then

• The sum function denoted by gf , is the function defined as )()( xgxfxgf

• The difference function denoted by gf , is the function defined as )()( xgxfxgf

• The product function denoted by fg , is the function defined as )()( xgxfxfg

• The quotient function denoted by g

f, is the function defined as

)(

)(

xg

xfx

g

f

• The composite function denoted by gf , is the function defined as ))(( xgfxgf

2

25

2

442

2

22

2

2

2

x

xx

x

xxx

x

xx

3. 22

2)2()()()(

x

x

xxxhxgxgh

4. 22

22

22

2

)(

)()(

x

x

xx

xx

x

xh

xfx

h

f

5. xxxfxgfxgf 222)()(

6.

2

23

2

2222

2

2

2

2

x

x

x

xx

x

x

x

xfxhfxhf

7. x

x

x

xxhxfhxfh

4

22

222

Note: In general, xfgxgf

Given the functions 2)( xxf , 2)( xxg and2

2)(

x

xxh , determine the following

1. 0gf 2. )1)(( hg 3. )5(gf

4. )2(fg 5. )1(

g

f 6. )1(gh

7. xgfh

Definition: A function f is said to be even if )()( xfxf , while a function is called odd if )()( xfxf .

Example

1. 2)( xxf , )()()( 22 xfxxxf . Thus f is an even function

2. 5)( 2 xxf , )(55)()( 22 xfxxxf . Thus f is an even function.

3. xxxf 5)( , )(1)()( 555 xfxxxxxxf . Thus f is an odd function.

4.1

)(

x

xxf ,

1)(

x

xxf . Note that )()( xfxf and )()( xfxf , thus f is neither odd nor even.

SOME TYPES OF FUNCTIONS

Constant function

The function cxf )( , where Rc , is a function that associates all real numbers with a fixed number c.

Domain: all real numbers. Range: c.

)(such that : xfYXf

Linear function

A function of the form, baxxf )( , where Rba , is called a linear function.

Domain: all real numbers Range: all real numbers

X Y

12)( xxf

3)( xxg

Quadratic functions

The quadratic function is of the form cbxaxxf 2)( , where Rcba ,, such that 0a .

xxxf 2)( 3)( 2 xxf

Square root function

xxf 3)( . Domain: 3| xxD Range: 0| yyR , the set of all non-negative real numbers.

xxg 3)( , 3| xxD Range: 0| yyR , the set of all non-positive real numbers.

Absolute Value Function s

12)( xxf Domain: all real numbers Range: 1| yy

Piece-wise Defined Functions

kkk axaxf

axaxf

axxf

xF

1

212

11

)(

)(

)(

)(

where, kaaa 21

The above function means that from the interval )1,( the value of the function follows the equation

12 xy and on the interval ),1[ the function follows the equation 3y .

REMARK

Given the graph, we can determine if the graph represents a function by a vertical line test. We note the

following: a graph represents a function if to each x in the domain there is only one value of y that corresponds

to that x. hence, by drawing a vertical line on any part of the graph, the line should intersect the graph in at most

one point, otherwise the graph does not represent a function.

Some illustrations:

Function Not a function

13

112)(

x

xxxf

Definition: A function is said to be one-to-one if and only if whenever yx, are in the domain of f, )()( yfxf

implies that yx .

Determine if the given function is one-to-one.

1. 23)( xxf . Suppose x and y are in the domain of f, and )()( yfxf , we have the following

2323)()( yxyfxf

yx

yx

33

Hence, the function f, is one-to-one.

2. 5)( 2 xxf 55)()( 22 yxyfxf

yx

yx

22

Now we see that when yx it does not necessarily follow that yx . Thus, the function f, is not one-to-one.

Consider the following functions 12)( xxf and 2

1)(

xxg . If we try to compute for the following

composite functions, observe the following:

xxx

xgxfgxfg

2

2

2

11212)()(

xxxx

fxgfxgf

1)1(1

2

12

2

1)()(

Definition: If a function f is one-to-one then there exists a function g such that xxfgxgf . We

usually denote the function xg as )(1 xf . We call )(1 xf the inverse of f.

Not a function

Alternative definition: If f is a one-to-one function, that is, the set of ordered pairs yx, is a function, and then

there is a function )(1 xf , read as “f inverse of x”. 1f is the set of ordered pairs xy, defined by )(1 yfx

if and only if xfy .

NOTE: )(

1)(1

xfxf

REMARK:

It follows from the definition of the inverse function, that the domain of 1f is the range of f and the range of 1f is the domain of f . This helps us find the range of a function.

Procedure to find the inverse of a function

STEP1: Interchange x and y in the equation )(xfy

STEP2: Solve the resulting equation for y in terms of x.

STEP3: Re-write the resulting equation as )(1 xfy

Example

1. 23)( xxf 23 xy Step1: 23 yx

Step2: 3

23223

xyyxyx

Step3: 3

2)(1 x

xf

2. 4

5)(

xxf

4

5

xy Step1:

4

5

yx

Step2: yxyx 5454

Step3: 54)(1 xxf

3. 2

1)(

x

xxf

2

1

x

xy Step1:

2

1

y

yx

Step2: 1212 yxxyyyx

1

21

211

21

x

xy

xxy

xyxy

Step3: 1

21)(1

x

xxf

Determine the domain and range of the following functions and find its inverse.

1. 35)( xxf 2. 12

)( x

xf

3. 21

)( x

xf 4. 1

)(

x

xxf

5. 12

5)(

x

xxf 6.

x

xxf

21

1)(

SAMPLE EXAM ON FUNCTIONS

GENERAL DIRECTIONS: Answer the following questions in the order given. Show all pertinent

solutions and enclose your answers whenever possible.

73)( xxf 12)( 2 xxg 12

1)(

x

xxh

Given the following functions above do the following:

1. Find

a) )3)(( tgf

b) )2)(( hg

c) )(1 xh

d) )(1 xf

2. Determine if the functions )()(),( xh and xgxf are odd, even or neither.

3. Find the domain and range of the functions f(x) and h(x).