Section 3.4

Objectives:

• Find function values

• Use the vertical line test

• Define increasing, decreasing and constant functions

•Interpret Domain and Range of a function Graphically and Algebraically

Function: A function f is a correspondence from a set D to a set E that assigns to each element x of D exactly one value ( element ) y of E

Graphical Illustration

E

x *

z *

w *

5 *

* f(w)

* f(x)

* f(z)

* f(5)

* 3* 4* - 9

D

f

f is a function

More illustrations….

x *

z *

w *

5 *

* f(w)

* f(x)

* f(z)

* f(5)

* 3* 4* - 9

D

E

f is not a function Why?

x in D has two values

x *

z *

w *

5 *

* f(w)

* f(x)

* f(z)

* f(5)

* 3* 4* - 9

D

E

f is not a function Why?

x in D has no values

Find function values

Example 1: Let f be the function with domain R such that f( x) = x2 for every x in R.

( i ) Find f ( -6 ), f ( ), f( a + b ), and f(a) + f(b) where a and b are real numbers.3

Solution: 3666 2 f

3332f

222 2 babababaf

22 babfaf Note: f ( a + b ) f( a ) + f ( b )

Vertical Line Testof functions

Vertical Line test: The graph of a set of points in a coordinate plane is the graph of a function if every vertical line intersects the graph in at most one point

Example: check if the following graphs represent a function or not

Function

Function

FunctionNot Function

Increasing, Decreasing and Constant Function

Terminology Definition Graphical Interpretation

f is increasing

over interval I

f(x1) < f(x2)

whenever

x1 < x2

f is decreasing

over interval I

f(x1) > f(x2)

whenever

x1 < x2

f is constant

over interval I

f(x1) = f(x2)

whenever

x1 = x2

x1 x2

f(x1)f(x2)

x

y

x1 x2

f(x1) f(x2)

x

y

x1 x2

f(x1) f(x2)

x

y

Example 1: Identify the interval(s) of the graph below where the function is

(a) Increasing

(b) Decreasing

Solution:

(a) Increasing ,20,

(b) Decreasing: 2,0

Example 2: Sketch the graph that is decreasing on ( ,- 3] and [ 0, ), increasing on [ -3 ,0 ],

f(-3) = 2 and f (2 ) = 0

Solution:

-3 0

decreasing increasing decreasing

Interpretation of Domain and Rangeof a function f

Domainis the

Set of all x where f is

well defined

Rangeis the set of all values

f( x )Where x is in the

domain

f

Graphical Approach toDomain and Range

Example 1: Find the natural domain and Range of the graph of the function f below

The function f represents f (x ) = x2. f is well defined everywhere in R. Therefore,

Domain = R ),(

Range

Domain

Every value of f is non-negative ( greater than or equal to 0. Therefore ,

Range = ),[ o

More illustrations of Domain and Range of a graph of a function f

These two graphs seem similar, but the domain and range are different

This graph does not end on both sides

Domain = ),(

Range = ),0[

This graph ends, it is also not defined at x = –2 and

well defined at x =2

Domain =

Range =

]2,2(

]4,0[

Class Exercise 1 Find the natural domain and range of the following graphs

Domain = Range = Domain =

Domain = Domain =

Range =

Range =

Range =

)2,2[ ]2,0[ ,R ]1,1[

,33,33,

,R

]25.5,75.0()25.2,75.6[

]3,75.0(3

Algebraic Approach to find theDomain of a function f

Example 1: Find the natural domain of the following functions

13

1024

102

133

1022

)(131

x

xxg

x

xxf

functionrootSquarexxf

functionLinearxxf

Solution:

( 1 ) f is a linear function. f is well-defined for all x. Therefore, Domain = R

( 2 ) f is a square root function. f is well defined when

0 10 2 x

5x Domain = ),5[

(3) f is well defined when

0 10 2 x 5x Domain = ),5(

(4) f is well defined when

0 10 2 x and 013 x

-5 3/1

Domain = ),3/1()3/1,5[

Do all the Homework assigned in the syllabus for

Section 3.4