**More on Functions and Their Graphs**

**Identify even or odd functions and recognize their symmetries.**

**Understand and use piecewise functions.**

Identify intervals on which a function increases, decreases, or is constant.

**Use graphs to locate relative**

maxima or minima.

maxima or minima.

A function is

increasing

when the y-value increases as the x-value increases

A function is

decreasing

, when the y-value decreases as the x-value increases

A

Constant

Function is a horizontal line

Example: Answer the following based on the graph below.

1. Identify the intervals in which f is increasing:

and (7,10)

2.Identify the intervals in which f is decreasing:

(2,7)

3. Identify the intervals on which f is constant:

(10,∞ )

∞

(0,2)

increasing

increasing

decreasing

decreasing

constant

Relative Maxima - greatest value

Relative Minima

- smallest value

Relative Maxima

(-0.67, 2.48)

Relative Minima

(2,-7)

Definition:

Relative Maximum

A function f(x) has a relative maximum value at x = a,

if f(a) is greater than any value immediately preceding or following.

Relative Minimum

A function f(x) has a relative minimum value at x = b,

if f(b) is less than any value immediately preceding or following.

Even Function

f(-x) = f(x)

f(x) = x + 2

2

f(-x) = (-x) + 2

= x + 2

2

2

They are equal, then f(x) is even.

Odd Function

f(-x) = - f(x)

f(x) = x + x

f(-x) = (-x) + (-x)

3

3

= -x - x

3

- f(x)

f(-x) = - f(x),

then f(x) is odd.

Example: Determine whether each of the following functions is even, odd, or neither:

a. f(x) =

x

x - 1

2

b. g(x) = x + x

4

2

c. h(x) = 2x - 3x - 4x + 4

3

2

a. f(x) =

x

x - 1

2

Find f(-x) by substituting -x for x in the given f(x).

f(-x) =

-x

(-x) - 1

2

=

-x

x - 1

2

= -

x

x - 1

2

(

)

f(-x) = - f(x)

f(x) is odd.

b. g(x) = x + x

4

2

Find g(-x) by substituting -x for x in the given g(x).

g(-x) = (-x) + (-x)

= x + x

4

4

2

2

g(-x) = g(x)

g(x) is even.

c. h(x) = 2x - 3x - 4x + 4

3

2

Find h(-x) by substituting -x for x in the given h(x).

h(-x) = 2(-x) - 3(-x) - 4(-x) + 4

3

2

= -2x - 3x + 4x + 4

3

2

h(-x) ≠ h(x) and h(-x) ≠ -h(x)

h(x) is neither even

nor odd.

Piecewise Functions

It is a function that is defined on a sequence of intervals.

when x is less than 1,

it gives -x

2

Example:

y = -x

when x<1

2

when x is 1, it gives 1

(1,1)

when x is greater than 1,

it gives x - 3

y = x - 3

when x>1

A solid dot means including.

An open dot means not including.

f(x)=

-x

1

x-3

2

**{**

if

if

if

x<1

x=1

x>1

And this is how it is written:

Example: Use the piecewise function g(x)

a. g(-1)

b. g(1)

g(x)=

2x + 1

-x + 5x

2

**{**

if

if

x<1

x≥1

to evaluate

a. g(-1)

Since x < 1, we use g(x) = 2x + 1. Substitute -1 for x in g(x) = 2x + 1.

g(

-1

) = 2(

-1

) + 1 = -2 + 1 = -1

b. g(1)

Since x = 1, we use g(x) = -x + 5x. Substitute 1 for x in g(x) = -x + 5x.

2

2

g(

1

) = -(

1

) + 5(

1

) = -1 + 5 = 4

Graphing a Piecewise Function

f(x)=

2x - 1

3

if

if

x≤1

x>1

{

Graph

for

.

f(x)=2x - 1

x≤1

x

f(x)=2x - 1

(x, f(x) )

1

0

-1

-2

f(

1

) = 2(

1

) - 1 = 1

f(

0

) = 2(

0

) - 1 = -1

f(

-1

) = 2(

-1

) - 1 = -3

f(

-2

) = 2(

-2

) - 1 = -5

(1,1)

(0,-1)

(-1,-3)

(-2,-5)

Graph

for

f(x)=3

x>1

.

x

f(x) = 3

(x, f(x) )

2

3

4

5

f(

2

) = 3

f(

3

) = 3

f(

4

) = 3

f(

5

) = 3

(2,3)

(3,3)

(4,3)

(5,3)

f(x) = 2x - 1

x≤1

f(x) = 2x - 1

x≤1

f(x) = 3; x>1

Find and Simplify a Function’s

Difference Quotient

Difference Quotient

This formula computes the slope of the secant line through two points on the graph of f.

f(x+h) - f(x)

h

h ≠ 0

Example: If

find and simplify

f(x) = 3x - 5x + 4

2

f(x+h) - f(x)

h

.

Find

by substituting

for

in

f(x+h)

(x+h)

x

f(x) = 3x - 5x + 4

2

.

f(

x+h

) = 3 (

x+h

) - 5(

x+h

) + 4

2

Simplify.

= 3 (x + 2hx + h ) - 5x - 5h + 4

2

2

= 3x + 6hx + 3h - 5x - 5h + 4

2

2

Use the result from part 1.

f(x+h) - f(x)

h

=

3x + 6hx + 3h - 5x - 5h + 4

2

2

- (3x - 5x + 4)

2

h

Simplify.

=

3x + 6hx + 3h - 5x - 5h + 4

2

2

- 3x + 5x - 4)

2

h

=

h

6hx + 3h - 5h

2

=

h

h(6x + 3h - 5)

Factor h.

f(x+h) - f(x)

6x + 3h - 5

h

=