CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 1. Functions and Graphs

Example B
Show thatf(x) = 4 x^3 −xis odd.
Solution:

f(−x) = 4 (−x)^3 −x

=− 4 x^3 +x

=−( 4 x^3 +x)

=−f(x)

Just like even functions are named, odd functions are named because negative signs don’t disappear and can always
be factored out of odd functions.
Example C
Identify whether the function is even, odd or neither and explain why.
f(x) = 4 x^3 −|x|
Solution:

f(−x) = 4 (−x)^3 −x

=− 4 x^3 −x

This does not seem to match eitherf(x) = 4 x^3 −|x|or−f(x) =− 4 x^3 +|x|. Therefore, this function is neither even
nor odd.
Note that this function is a difference of an odd function and an even function. This should be a clue that the resulting
function is neither even nor odd.
Concept Problem Revisited
Even and odd functions describe different types of symmetry, but both derive their name from the properties of
exponents. A negative number raised to an even number will always be positive. A negative number raised to an odd
number will always be negative.

Vocabulary

Aneven functionmeansf(−x) =f(x). Even functions havereflection symmetryacross the linex=0.
Anodd functionmeansf(−x) =−f(x). Odd functions haverotation symmetryabout the origin.

Guided Practice

1. Which of the basic functions are even, which are odd and which are neither?


2. Supposeh(x)is an even function andg(x)is an odd function.f(x) =h(x)+g(x).Isf(x)even or odd?


3. Determine whether the following function is even, odd, or neither.
   f(x) =x(x^2 − 1 )(x^4 + 1 )