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More on Functions and Their Graphs

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Juan Jorrin

on 16 February 2017

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Transcript of More on Functions and Their Graphs

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.

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
=
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