More on Functions & Graphs
2.2
JMerrill, 2007
Contributions by DDillon
Revised 2008
Review

Find:

Domain


Range


[-5, 4]
f(-1)


[-1, 4)
f(-1) = -5
f(2)

f(2) = 4
Difference Quotient

One of the basic definitions in calculus uses
the difference quotient ratio:
f(x  h)  f(x)
h

It applies to average rate of change.
Difference Quotient

For f(x) = x2 – 4x + 7, find
f(x  h)  f(x)
h
(x  h)2  4(x  h)  7  (x2  4x  7)
h
x2  2xh  h2  4x  4h  7  x2  4x  7
h
h(2x  h  4)
2xh  h2  4h

h
h
 2x  h  4
Difference Quotient You Do

Given f(x) = 3x – 1, find

3
f(x  h)  f(x)
h
A piecewise-defined function is composed of two or
more functions.
f(x) =
3 + x, x < 0
x2 + 1, x  0
Use when the value of x is less than 0.
Use when the value of x is greater or
equal to 0.
y
open circle
closed circle
(0 is not included.)
(0 is included.)
x
4
-4
Evaluating
A Piecewise-Defined Function
x2  1, x  0
f(x)  
x  1, x  0

Evaluate the function
x = -1 and x = 0

When x = -1, that is less than 0, so you only
use the top function


f(-1) = (-1)2 + 1 = 2
When x = 0, use the bottom function

f(0) = 0 – 1 = -1
when
You Do




Solve f ( x )  2x  1,
2x  2,
A. f(-1)
B. f(0)
C. f(2)
f(-1) = -1
f(0) = 2
f(2) = 6
x0
x 0
Where is this function increasing?
Where is it decreasing?
The graph of y = f (x):
(–3, 6)
y
• increases on (– ∞, –3),
• decreases on (–3, 3),
• increases on (3, ∞).
x
(3, – 4)
A function f is:
• increasing on an interval if, for any x1 and x2 in the
interval, x1 < x2 implies f (x1) < f (x2),
P206
• decreasing on an interval if, for any x1 and x2 in the
interval, x1 < x2 implies f (x1) > f (x2),
• constant on an interval if, for any x1 and x2 in the
interval, f (x1) = f (x2).
y
(–3, 6)
The graph of y = f (x):
• increases on (– ∞, –3),
• decreases on (–3, 3),
• increases on (3, ∞).
x
(3, – 4)
Function Extrema
(or local)
(or local)
Find Extrema and Intervals of
Increasing and Decreasing Behavior.
y = x3 – 3x
Relative max exists at -1.
Relative max = 2
Relative min is exists at 1.
Relative min = -2
Increasing:  -, -1
Decreasing:  1,1
Increasing: 1,  
Application
During a 24-hour period, the temperature y (in degrees
Fahrenheit) of a certain city can be approximated by the
model y = 0.026x3 – 1.03x2 + 10.2x + 34, 0 ≤ x ≤ 24,
where x represents the time of day, with x = 0
corresponding to 6 AM. Approximate the maximum and
minimum temperatures during this 24-hour period.
Maximum: about 64°F (at 12:36 PM)
Minimum: about 34°F (at 1:48 AM)
A Function f is even if for each x in the domain of f,
f (– x) = f (x).
f (x) = x2
f (– x) = (– x)2 = x2
If you get the same thing you started with, it is an even function
f (x) = x2 is an even function.
A Function f is even if for each x in the domain of f,
f (– x) = f (x).
f (x) =
x2
y
An even function is
symmetric about the
y-axis.
x
A Function f is odd if for each x in the domain of f,
f (– x) = – f (x).
f (x) = x3
f (– x) = (– x)3 = –x3
If all terms change signs the function is odd.
f (x) = x3 is an odd function.
A Function f is odd if for each x in the domain of f,
f (– x) = – f (x).
f (x) = x3
y
An odd function is
symmetric with
respect to the origin.
x
Summary of Even and Odd
Functions & Symmetry
1. Replace x with –x
2. Simplify
3. If nothing changes, the
function is even. If
everything changes, the
function is odd.
Even, Odd, or Neither?
f(x) = x3 + 2
Check f(-x)
f(-x) = (-x)3 + 2
f(-x) = -x3 + 2
Not even, because not equal to f(x).
Not odd, because not equal to –f(x).
This function is neither even nor odd.