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Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Even and Odd Functions

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Even and Odd Functions

A function f(x)f\left(x\right) is even if, for all x in its domain,
f(x)=f(−x)\boxed{\quad f\left(x\right)=f\left(-x\right)\quad}

A function f(x)f\left(x\right) is odd if, for all x in its domain if
f(−x)=−f(x)\boxed{\quad f \left(-x\right)=-f\left(x\right)\quad}


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Determining if a Function is Even or Odd
  • Work out f(−x)f\left(-x\right) first, and compare it to f(x)f(x). If they are the same, the function is even.
  • If not, find −f(x)-f\left(x\right) next and compare it to f(−x)f\left(-x\right). If they are the same, the function is odd. If not, the function is neither even nor odd.

Properties

  • A function can be even, odd, or neither.
  • The sum of two even functions is even. The sum of two odd functions is odd.
  • The sum of an even and an odd function is neither even nor odd (unless one of them is zero).
  • The product of two even or two odd functions is even.
  • The product of an even and an odd function is odd.
  • The reciprocal of an even/odd function is even/odd.

Examples:
  • The function y=x2y=x^2 is even because for any x, we have: f(x)=x2=(−x)2=f(−x)f\left(x\right)=x^2=\left(-x\right)^2=f\left(-x\right)
  • The function y=x3y=x^3 is odd because for any x, we have: f(−x)=(−x)3=−x3=−f(x)f\left(-x\right)=\left(-x\right)^3=-x^3=-f\left(x\right)
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Example: Even and Odd Functions

Determine if the following function is even, odd, or neither: f(x)=x5−2x3+8f(x)=x^5-2x^3+8

Check f(−x)f\left(-x\right):

f(−x)=(−x)5−2(−x)3+8=−x5+2x3+8f\left(-x\right)=\left(-x\right)^5-2\left(-x\right)^3+8=-x^5+2x^3+8
Since f(−x)≠f(x)f\left(-x\right)\ne f(x), the function is not even.


Check −f(x)-f(x):

−f(x)=−x5+2x3−8-f(x)=-x^5+2x^3-8
Since f(−x)≠−f(x)f\left(-x\right)\ne-f(x), the function is not odd.


Therefore the function is neither even nor odd.

Extra Practice
Properties of Functions
Which of the following statements is/are true about the functions f(x)=0f\left(x\right)=0, g(x)=x3 cos⁡xg\left(x\right)=x^3\ \cos x, and h(x)=esin⁡xh\left(x\right)=e^{\sin x}?
i. Exactly two of the functions are even.
ii. Exactly two of the functions are odd.
Combining Families of Functions
Practice: Combining Families of Functions
Is the function f(x)=x2x2−6f(x)=\dfrac{x^2}{x^2-6} even, odd, or neither?
Combining Families of Functions
Practice: Combining Families of Functions
Is the function f(x)=xx2−1f(x)=x\sqrt{x^2-1} even, odd, or neither?
Functions & Graph Properties
Practice: Functions & Graph Properties
True or False:
The function y=−2x3+5y=-2x^3+5 is odd.
Functions & Graph Properties
Practice: Functions & Graph Properties
True or False:
The function y=xx2−1y=\dfrac{x}{x^2-1} is odd.
Properties of Functions
Which of the following statements is/are true about the functions f(x)=0f\left(x\right)=0, g(x)=x3 cos⁡xg\left(x\right)=x^3\ \cos x, and h(x)=esin⁡xh\left(x\right)=e^{\sin x}?
i. Exactly two of the functions are even.
ii. Exactly two of the functions are odd.