1.3- Functions and their Graphs

Example

Solutiona.f (-l) = 2. b.f (1) = 4.

Use the graph of the function f to answer the following questions:

• What are the function values f (-1) and f (1)?• What is the domain of f (x)?• What is the range of f (x)?

-5 -4 -3 -2 -1 1 2 3 4 5

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SolutionThe domain of f is

{ x | -3 < x < 6} or the interval (-3, 6].

-5 -4 -3 -2 -1 1 2 3 4 5

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Example cont.Use the graph of the function f to answer the

following questions.• What are the function values f (-1) and f (1)?• What is the domain of f (x)?• What is the range of f (x)?

SolutionThe range of f is { y | -4 < y < 4} or

the interval (-4, 4]. -5 -4 -3 -2 -1 1 2 3 4 5

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Use the graph of the function f to answer the following questions.

• What are the function values f (-1) and f (1)?• What is the domain of f (x)?• What is the range of f (x)?

Example cont.

The Vertical Line Test for Functions

• If any vertical line intersects a graph in more than one point, the graph does not define y as

a function of x.

Text Example

Solution y is a function of x for the graphs in (b) and (c).

Use the vertical line test to identify graphs in which y is a function of x.

x

y

a.

x

y

b.

x

y

c.

x

y

d.

x

y a.

y is not a function since 2 values of y correspond to an x-value.

x

y b.

y is a function of x.

x

yc.

y is a function of x.

x

yd.

y is not a function since 2 values of y correspond to an x-value.

Ex. Determine the domain of the function:

( ) 4f x x

4 0x

4 x 4x

( , 4]

Increasing, Decreasing, and Constant Functions

Constant

(x1, f (x1))

(x2, f (x2))

Increasing

(x1, f (x1))

(x2, f (x2))

Decreasing

(x1, f (x1))

(x2, f (x2))

Solution

a. Decreasing on the interval (-oo, 0), increasing on the interval (0, 2), and decreasing on the interval (2, oo).

Describe the increasing, decreasing, or constant behavior of each function whose graph is shown.

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-5 -4 -3 -2 -1 1 2 3 4 5

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a. b.

Example

Solution

b. Constant on the interval (-oo, 0).

Increasing on the interval (0, oo).

Describe the increasing, decreasing, or constant behavior of each function whose graph is shown.

-5 -4 -3 -2 -1 1 2 3 4 5

5

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-5 -4 -3 -2 -1 1 2 3 4 5

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a. b.

Example cont.

Definitions of Relative Maximum and Relative Minimum

1. A function value f(a) is a relative maximum if it is a “peak” in the graph.

2. A function value f(b) is a relative minimum of f if is a “valley in the graph.

x

y

Find the maximum and minimum points:

Ex. Graph the function:2 3, 1

( )4, 1

x xf x

x x

Definition of Even and Odd Functions

The function f is an even function if:f (-x) = f (x) for all x in the domain of f.

The function f is an odd function if:f (-x) = -f (x) for all x in the domain of f.

Example:Identify the function as even, odd, or neither:

Solution:We use the given function’s equation to find f(-x):

2( ) 3 2f x x

2( ) 3( ) 2f x x 23 2x ( ) ( )f x f x

So this is an EVEN function!

Example:Identify the function as even, odd, or neither:

3( ) 6f x x x Replace x with -x: 3( ) ( ) 6( )f x x x

3( ) 6f x x x ( )f x

( ) ( )f x f x

So this is an ODD function!

Even Functions and y-Axis Symmetry• The graph of an EVEN function in which f (-x) = f (x) is:

symmetric with respect to the y-axis.

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y

Odd Functions and Origin Symmetry• The graph of an odd function in which f (-x) = - f (x) is: symmetric with respect to the origin.

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y