Odd and even functions are two functions with important features. An even function exhibits symmetry about the *y*-axis. On the other hand, an odd function has 180° rotational symmetry about the origin. It is possible to determine whether a function is odd or even using algebraic methods.

In this article, we will learn all about even and odd functions. We will look at their graphs and some important characteristics. We will also learn how to determine if a function is even or odd.

## Characteristics and graph of the even function

Even functions have the main characteristic that they are symmetric about the *y*-axis. This means that if we fold the graph of an even function on the *y*-axis, we get two equal parts of the graph.

An important even function is the function $latex f(x)=x^2$, which has the following graph:

Other examples of even functions are the following functions:

- $latex f(x)=\cos(x)$
- $latex f(x)=x^4$
- $latex f(x)=\sin^2(x)$

### Formula to test if a function is even

To determine if a function is even or not, we can use the following formula:

$latex f(-x)=f(x)$

This means that if when replacing the values of *x* in the function with –*x* we obtain the original function, then the function is even.

## Characteristics and graph of the odd function

The main characteristic of odd functions is that they have 180° rotational symmetry about the origin. This means that if we rotate the graph 180° about the point (0, 0), the graph will not change.

An important odd function is the function $latex f(x)=x^3$, which has the following graph:

Other examples of odd functions are the following functions:

- $latex f(x)=\sin(x)$
- $latex f(x)=x^5$
- $latex f(x)=x^7$

### Formula to test if a function is odd

We can determine if a function is even or odd using the following formula:

$latex f(-x)=-f(x)$

This means that if when replacing the *x* values in the function with –*x* we obtain the original function multiplied by -1, then the function is odd.

## Examples of even and odd functions

In the following examples, we can apply what we have learned about the range of a function. Each of the examples has a detailed solution using the graph of the function.

### EXAMPLE 1

Determine whether the function $latex f(x)={{x}^2}+2$ is even or odd.

**Solution:** For a function to be even, the condition $latex f(-x)=f(x)$ must be true. Therefore, let’s check this:

$latex f(-x)= (-x)^2+2$

$latex f(-x)= x^2+2$

We see that we got the original function. So the function is even. The following is its graph and we can verify that it is even, since it is symmetric with respect to the y-axis.

### EXAMPLE 2

Is the function $latex f(x)=3x^2-|x|$ even or odd?

**Solution:** We can determine if the function is even or odd by testing $latex f(-x)=f(x)$ for even functions and $latex f(-x)=-f(x)$ for odd functions.

Starting with the test for even functions, we have:

$latex f(-x)=3(-x)^2-|-x|$

$latex f(-x)=3x^2-|x|$

We see that the expression obtained is equal to the original function $latex f(x)$, so the function is even.

Since the function is even, we no longer have to test for odd functions.

### EXAMPLE 3

Determine whether the function $latex f(x)=\frac{3}{x}+2x$ is odd or even.

**Solution:** To solve this problem, we can apply the test for even functions and then perform the test for odd functions.

If the function is even, we must have $latex f(-x)=f(x)$. Then, we have:

$latex f(-x)=\frac{3}{-x}+2(-x)$

$latex f(-x)=-\frac{3}{x}-2x$

This function is not even because the expression obtained is not equal to the original function $latex f(x)$.

Looking at the expression obtained, we see that it is equal to $latex -f(x)$. This means that the function is $latex because f(-x)=-f(x)$.

### EXAMPLE 4

Is the function $latex f(x)=(x^3-5)^2$ even, odd, or neither?

**Solution:** Using the test for even functions, $latex f(-x)=f(x)$, we have:

$latex f(-x)=((-x)^3-5)^2$

$latex f(-x)=(-x^3-5)^2$

The expression we got is not equal to $latex f(x)$, so the function is not even. Therefore, we continue with the test for odd functions.

To check if the function is odd, we must have $latex f(-x)=-f(x)$. However, the function is also not odd, since the expression obtained above is not equal to $latex -f(x)$.

**Note:** $latex -f(x)$ would be equal to $latex -(x^3-5)^2$.

## See also

Interested in learning more about algebraic functions? Take a look at these pages:

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