FUNCTIONS

OBJECTIVES:• define functions;•distinguish between dependent and independent variables;•represent functions in different ways; and•evaluate functions•define domain and range of a function; and•determine the domain and range of a function

DEFINITION: FUNCTION

• A function is a special relation such that every first element is paired to a unique second element.

• It is a set of ordered pairs with no two pairs having the same first element.

• A function is a correspondence from a set X of real numbers x to a set Y of real numbers y, where the number y is unique for a specific value of x.

xy sin=13 += xy

One-to-one and many-to-one functions

Each value of x maps to only one value of y . . .

Consider the following graphs

Each value of x maps to only one value of y . . .

BUT many other x values map to that y.

and each y is mapped from only one x.

and

Functions

One-to-one and many-to-one functions

is an example of a one-to-one function

13 += xy is an example of a many-to-one function

xy sin=

xy sin=13 += xy

Consider the following graphs

and

Functions

One-to-many is NOT a function. It is just a relation. Thus a function is a relation but not all relation is a function.

In order to have a function, there must be one value of the dependent variable (y) for each value of the independent variable (x). Or, there could also be two or more independent variables (x) for every dependent variable (y). These correspondences are called one-to-one correspondence and many-to-one correspondence, respectively. Therefore, a function is a set of ordered pairs of numbers (x, y) in which no two distinct ordered pairs have the same first number.

Ways of Expressing a function

5. Mapping2. Tabular form

3. Equation

4. Graph1. Set notation

.

Example: Express the function y = 2x;x= 0,1,2,3 in 5 ways.

1. Set notation (a) S = { ( 0, 0) , ( 1, 2 ) , ( 2, 4 ), ( 3, 6) }

or (b) S = { (x , y) such that y = 2x, x = 0, 1, 2,

3 }2. Tabular

form x 0 1 2 3

y 0 2 4 6

3. Equation: y = 2x

4. Graph

y

x5-4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2-5

-3

●

●

●1 2

2 4

63

0 0

x y

5. Mapping

EXAMPLE:Determine whether or not each of the following sets represents a function:1.A = {(-1, -1), (10, 0), (2, -3), (-4, -1)}

2. B = {(2, a), (2, -a), (2, 2a), (3, a2)}

3. C = {(a, b)| a and b are integers and a = b2}

4. D = {(a, b)| a and b are positive integers and a = b2}

5. ( ){ }4xy|y,xE 2 −==

There are more than one element as the first component of the ordered pair with the same second component namely (-1, -1) and (-4, -1), called a many-to-one correspondence. One-to-many correspondence is a not function but many-to-one correspondence is a function.

There exists one-to-many correspondence namely, (2, a), (2, -a) and (2, 2a).

SOLUTIONS:

1. A is a function.

2. B is a not a function.

3. C is not a function. There exists a one-to-many correspondence in C such as (1, 1) and (1, -1), (4, 2) and (4, -2), (9, 3) and (9, -3), etc.

4. D is a function. The ordered pairs with negative values in solution

c above are no longer elements of C since a and b are given as positive integers. Therefore, one-to-many correspondence does not exist anymore in set D.

5. E is not a function Because for every value of x, y will have two values.

OTHER EXAMPLES:

Determine whether or not each of the following sets represents a function:a) S = { ( 4, 7 ), ( 5, 8 ), ( 6, 9 ), ( 7, 10 ), ( 8, 11 ) }

b) S = { ( x , y ) s. t. y = | x | ; x ∈ R }

c) y = x 2

– 5 d) | y | = x

2x

x2y

+=e)

1xy +=f)

DEFINITION: FUNCTION NOTATION

• Letters like f , g , h, F,G,H and the likes are used to designate functions.

• When we use f as a function, then for each x in the domain of f , f ( x ) denotes the image of x under f .

• The notation f ( x ) is read as “ f of x ”.

EXAMPLE: Evaluate each function value.

1. If f ( x ) = x + 9 , what is the value of f ( x 2 ) ?

2. If g ( x ) = 2x – 12 , what is the value of g (– 2 )?

3. If h ( x ) = x 2 + 5 , find h ( x + 1 ).

4.If f(x) = x – 2 and g(x) = 2x2 – 3 x – 5 , Find: a) f(g(x)) b) g(f(x))

[ ] [ ] 222 )(,)(),1(),(),0(),3( hafafafafff ++5. If find each of the following1)( 2 −= xxf

h

)x(g)hx(g −+6. Find (a) g(2 + h), (b) g(x + h), (c)

where h≠ 0 if .

1x

x)x(g

+=

7. Given that show that

.

,1

)(x

xF =kxx

kxFkxF

+−=−+2

)()(

323 4),( yxyxyxf ++= ),(),( 3 yxfaayaxf =8. If , show that

,),(vu

vuvuf

+−= ( )vuf

vuf ,

1,1 −

9. If find

DEFINITION: DEFINITION: Domain and RangeDomain and Range

All the possible values of x is called the domain and all the possible values of y is called the range. In a set of ordered pairs, the set of first elements and second elements of ordered pairs is the domain and range, respectively.

Example: Identify the domain and range of the following functions.

1.) S = { ( 4, 7 ),( 5, 8 ),( 6, 9 ),( 7, 10 ),( 8, 11 ) }

Answer : D: { 4,5,6,7,8} R:{7,8,9,10,11}

2.) S = { ( x , y ) s. t. y = | x | ; x ∈ R }Answer: D: all real nos. R: all real nos. > 0

3) y = x 2 – 5

Answer. D: all real nos. R: all real nos. > -5

),( +∞−∞ ),0[ +∞

),( +∞−∞ ),5[ +∞−

2x

x2y

+=4.

Answer: D: all real nos. except -2

R: all real nos. except 2

1xy +=5. Answer : D: all real nos. > –1 R: all real nos. > 0

3x

x3y

−−=6.

Answer:D: all real nos. <3R: all real nos. <0

2except),(:D −+∞−∞ 2),(: ++∞−∞ exceptR

),1[:D +∞− ),0[:R +∞

)3,(:D −∞ ( )0,:R ∞−

From the above examples, you can draw conclusions and formulate the following theorems on the domain determination of functions.

Theorem 1. The domain of a polynomial function is the set of all real numbers or (-∞, +∞).

Theorem 3. A rational function f is a ratio of two polynomials: The domain of a rational function consists of all values of x such that the denominator is not equal to zero

where P and Q are polynomials.

)x(Q

)x(P)x(f =

Theorem 2. The domain of is the set of all real numbers satisfying the inequality f(x) ≥ 0 if n is even integer and the set of all real numbers if n is odd integer.

n )x(f

1.An algebraic function is the result when the constant function, (f(x) = k, k is constant) and the identity function (g(x) = x) are put together by using a combination of any four operations, that is, addition, subtraction, multiplication, division, and raising to powers and extraction of roots.

KINDS OF FUNCTIONS:

Example: f(x) = 5x – 4, 4x7x2

x)x(g

2 −+=

Generally, functions which are not classified as algebraic function are considered as transcendental functions namely the exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic andinverse hyperbolic functions.

73

1 +−

=y

x

A. Which of the following represents a function?1.A = {(2, -3), (1, 0), (0, 0), (-1, -1)}2.B = {(a, b)|b = ea}3.C = {(x, y)| y = 2x + 1}4. 5. E = {(x, y)|y = (x -1)2 + 2}6. F = {(x, y)|x = (y+1)3 – 2}7. G = {(x, y)|x2 + y2 = }8. H = {(x, y)|x ≤ y}9. I = {(x, y)| |x| + |y| = 1}10. J = {(x, y)|x is positive integer and

EXERCISES:

( ){ }21|, abbaD −==

B. Given the function f defined by f(x) = 2x2 + 3x – 1, find:a. f(0) f. f(3 – x2)b. f(1/2) g. f(2x3)c. f(-3) h. f(x) + f(h)d.f(k + 1) i. [f(x)]2 – [f(2)]2

e. f(h – 1) j. 0h;h

)x(f)hx(f ≠−+

,32)( += xxFC. Given find

0;)()(

.5

,2

1.4

),32(.3

),4(.2

),1(.1

≠−+

+

−

hh

xFhxF

F

xF

F

F

EXERCISES:

Find the domain and range of the following functions:

22

2

2

4:.6

1

12)(.5

3)(.4

21)(.3

1)(.2

34)(.1

xyg

x

xxxf

xxh

xxG

xxF

xxf

+=−

+−=

+=

−=+=

−=

( )( )( )( )312

943:.10

4:.9

3

1

3

)(.8

12

1)(.7

2

22

+−+−−+=

−=

−−

=

+−

=

xxx

xxxyH

xyg

xf

x

xxF

if

if

3

3

≥<x

x

if

if

if

2

21

1

≥<<

≤

x

x

x

Exercises: Identify the domain and range of the

following functions. 1. {(x,y) | y = x 2 – 4 }

8. y = (x 2 – 3) 2

−=

x2

x3y)y,x(4.

{ } 3),( xyyx =2.

{ } 9),( −= xyyx3.

{ }4x3xy)y,x( 2 −−=5.

y = | x – 7 |6.

7. y = 25 – x 2

x

5x3y

+=9.

5x

25xy

2

+−=10.