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Section 3.1
Power Functions
Objectives:
1. To define and evaluate power
functions.
2. To define even and odd functions.
3. To graph power functions and
identify the domain and range.
Definition
Power function A function of
the form f(x) = Cxn where C, n
 {real numbers}.
Notice that the definition
includes functions in which n
is rational or irrational. We will
only be looking at functions
with positive integral
exponents. This means they
will be polynomial functions
and require you to use your
knowledge of polynomials.
Power functions differ from
polynomial functions in that
they only have one term, and
exponents can be any real
number. Polynomial functions
can have only non-negative
integer exponents.
EXAMPLE 1 For f(x) = -2x4 evaluate
f(-1), f(0), f(1/2), f(3). Find the degree, the
domain, the range, and graph the
function.
f(-1) = -2(-1)4 = -2(1) = -2
f(0) = -2(0)4 = -2(0) = 0
f(1/2) = -2(1/2)4 = -2(1/16) = -1/8
f(3) = -2(3)4 = -2(81) = -162
EXAMPLE 1 For f(x) = -2x4 evaluate
f(-1), f(0), f(1/2), f(3). Find the degree, the
domain, the range, and graph the
function.
The degree is 4
D = {real numbers}
R = {y|y  0}
EXAMPLE 1 For f(x) = -2x4 evaluate
f(-1), f(0), f(1/2), f(3). Find the degree, the
domain, the range, and graph the
function.
2
1
-3 -2 -1-1
-2
-3
-4
-5
1 2
3
EXAMPLE 2 Graph g(x) = x3. Give
the domain and range.
D = {real numbers}
6
R = {real numbers}
4
2
-4
-2
2
-2
-4
4
All equation of the form f(x) =
Cxn are functions (passing the
vertical line test) with domain
D = {real numbers}. A parabola
also always has line symmetry.
Practice: Graph f(x) = -x4.
Give the domain and range.
0 = - x4
Roots: x = 0, multiplicity 4
f(1) = - (1)4 = - 1
f(-1) = - (-1)4 = - 1
y-axis symmetry (line symmetry)
y
x
Domain: all real numbers
Range: (-, 0]
Practice: Graph f(x) = 2x3.
Give the domain and range.
0 = 2x3
Roots: x = 0 multiplicity 3
f(1) = 2(1)3= 2
f(-1) = 2(-1)3 = -2
origin symmetry (point symmetry)
y
x
Domain: all real numbers
Range: all real numbers
Definition
Even function A function is
even if and only if f(x) = f(-x),
 x  Df.
Definition
Odd function A function is
odd if and only if f(-x) = -f(x), 
x  Df.
Power functions of even
degree are even functions and
power functions of odd degree
are odd functions.
One special function is the
identity function, y = x. This
is a power function of degree
1. The identity function is an
odd function.
EXAMPLE 3 Determine whether the
following functions are even, odd, or
neither.
f(x) = x3
f(-x) = (-x)3 = -x3
-f(x) = -(x3) = -x3
Since f(-x) = -f(x) the function is odd.
EXAMPLE 3 Determine whether the
following functions are even, odd, or
neither.
g(x) = x4 + x2
g(-x) = (-x)4 + (-x)2 = x4 + x2
-g(x) = -(x4 + x2) = -x4 – x2
Since g(x) = g(-x) the function is even.
EXAMPLE 3 Determine whether the
following functions are even, odd, or
neither.
h(x) = x2 + 2x + 5
h(-x) = (-x)2 + 2(-x) + 5 = x2 – 2x + 5
-h(x) = -(x2 + 2x + 5) = -x2 – 2x – 5
Since h(x)  h(-x)  -h(x) the function is
neither even nor odd.
Practice: Classify the function f(x) =
2x3 – 5.
1. Even
2. Odd
3. Neither
Practice: Identify the domain of the
function f(x) = 2x3 – 5.
1. {x|x  real numbers}
2. {y|y  -5}
3. {y|y  real numbers}
4. None of these
Practice: Determine whether
the following functions are
even, odd, or neither.
f(x) = 4x5 + 2x3 – x
f(-x) = - 4x5 - 2x3 + x
-f(x) = -(4x5 + 2x3 - x) = - 4x5 - 2x3 + x
f(-x)= - f(x)
 f(x) is odd.
Practice: Determine whether
the following functions are
even, odd, or neither.
g(x) = 3x4 - 5x2
4 -45x
2
2
g(-x) = 3x
3(-x)
- 5(-x)
g(x) = g(-x)
Therefore, g(x) is even.
Practice: Determine whether
the following functions are
even, odd, or neither.
h(x) = x3 - x2 + x - 1
2-x
2+
h(-x) = -(-x)
x33- -x(-x)
-1(- x) - 1
-h(x) = - x(x33+- x2 -+xx+- 11)
h(x)  h(-x)
h(-x)  -h(x)
Therefore, h(x) is neither even nor
odd.
Homework:
pp. 109-110
►A. Exercises
Graph each power function. Give the
domain and range of each and classify as
even or odd.
5. f(x) = -1/4x4
►A. Exercises
Graph each power function. Give the
domain and range of each and classify as
even or odd.
7. y = 5/12x24
►B. Exercises
Evaluate.
13. f(-3) for f(x) = -2x3
►B. Exercises
Evaluate.
15. f(-17.95) for f(x) = -2.5x16
►B. Exercises
For f(x) = Cxn.
16. Find f(1)
►B. Exercises
For f(x) = Cxn.
n
17. Find f( 3)
►B. Exercises
For f(x) = Cxn.
18. Find all zeros.
►B. Exercises
For f(x) = Cxn.
19. What is the multiplicity of the zero?
►B. Exercises
For f(x) = Cxn.
20. Give the domain of f(x).
►B. Exercises
For f(x) = Cxn.
21. Give the range of f(x) if n is odd.
►B. Exercises
For f(x) = Cxn.
22. If n is even, on what does the range of
f(x) depend?
►B. Exercises
For f(x) = Cxn.
23. Give the range of f(x) if n is even.
■ Cumulative Review
28. In ABC, find C, given a = 47, b = 63,
and c = 82.
■ Cumulative Review
29. Is the relation in the graph a function?
■ Cumulative Review
30. If sin x = 0.3, find csc x, cos (90 – x),
and sec (90 – x).
■ Cumulative Review
31. Solve 2x² – 5x + 7 = x(x + 1).
■ Cumulative Review
32. Find the slope of the line joining (2, 7)
to (4, -5).