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# 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 $3{x}^{3}+4x$ . 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:

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

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

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

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

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

b. Define a function $f\left(x\right)=\mathrm{log}\left({x}^{2}\right)$ . 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:

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

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

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

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

## Flashcards covering the Even and Odd Functions

Algebra II Flashcards

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