Even and Odd Functions

You've probably explored even and odd numbers in the past, but what about even and odd functions? Yes, they exist. No, determining whether a function is even or odd isn't as easy as checking whether the first term is an even or odd number. In this article, we'll see what makes a function even or odd and what that means when it's graphed on the Cartesian plane. Let's get started!

Even and odd functions: the even

A function is considered even if, for each x in the domain $f,f\left(-x\right)=f\left(x\right)$ . For instance:

$f\left(-x\right)=x^2$

We can test whether this is an even function by subbing -x for x and seeing if we get the same equation.

$f\left(-x\right)=\left(-x\right){\left(-x\right)}^2=x^2=f\left(x\right)$

This is indeed an even function. When graphed, even functions have reflective symmetry across the y-axis as illustrated below:

Remember that a function is only even if every potential value of x satisfies the criteria. We cannot find a single value for x where $f\left(-x\right)=-f\left(x\right)$ and call the function even.

Even and odd functions: the odd

A function is considered odd if, for each x in the domain of f, $f\left(-x\right)=-f\left(x\right)$ . Again, we can test it algebraically:

$f\left(x\right)=x^3$

$f\left(-x\right)=\left(-x\right){\left(-x\right)}^3=-x^3=-f\left(x\right)$

$f\left(x\right)=x^3$ is indeed an odd function. When graphed, odd functions have 180° rotational symmetry around the origin as shown:

The vast majority of functions won't satisfy the criteria for even or odd functions and are therefore neither. Never assume that a function must be even or odd.

Practice questions on even and odd functions

a. Define a function as $3x^3+4x$ . Is this function even, odd, or neither?

We need to find f(-x) to determine its symmetry. If $f\left(-x\right)=f\left(x\right)$ , the function is even. If $f\left(-x\right)=-f\left(x\right)$ , the function is odd. Otherwise, the function is neither even nor odd. Let's substitute -x for x:

$f\left(-x\right)=3{\left(-x\right)}^3+4\left(-x\right)$

We can simplify this using the Power of a Product Property:

$f\left(-x\right)=3{\left(-x\right)}^3+4\left(-x\right)$

$f\left(-x\right)=3\left(-1\right){\left(x\right)}^3+4\left(-x\right)$

$f\left(-x\right)=-3x^3-4x$

Comparing this with $f\left(x\right)=3x^3+4x$ , we find $f\left(-x\right)=-\left(3x^3+4x\right)=-f\left(x\right)$

Since $f\left(-x\right)=-f\left(x\right)$ , the function is odd.

b. Define a function $f\left(x\right)=\log \left(x^2\right)$ . Is this function even, odd, or neither?

We're working with logarithms this time, but that doesn't change what we have to do. First, we need to determine what $f\left(-x\right)$ is by subbing -x in for x:

$f\left(-x\right)=\log \left(-{\left(-x\right)}^2\right)$

$f\left(-x\right)=\log {\left(x\right)}^2$

$f\left(-x\right)=f\left(x\right)$

Since $f\left(-x\right)=f\left(x\right)$ , this is an even function.

Topics related to the Even and Odd Functions

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Parent Graphs

Degree (of a Polynomial)

Flashcards covering the Even and Odd Functions

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College Algebra Flashcards

Practice tests covering the Even and Odd Functions

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College Algebra Diagnostic Tests

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