There are all types of ways to describe numbers. We have positive numbers and negative numbers. There are rational and irrational numbers. You get the picture.

Most of the ways we use to describe numbers can also be used to describe functions. There are rational numbers and rational functions, and they both share similar characteristics. Even and odd, while used to describe numbers, can also describe functions.

**Even functions are functions that remain the same when all the x’s are replaced with a negative x. Odd functions are functions that become the negative versions of themselves when you replace all x’s with a negative x.**

We’ll get more into the specifics later. But let’s dive deeper into what both functions are, and what their graphs look like.

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## What are even functions?

We know that the domain of a function consists of all the x values. The range of a function consists of all the y values.

An even function occurs when you can replace all the x’s in the function with a negative x and retain the original function. When you put negative values in, you keep the original function.

**Even function are functions where f(-x) = f(x)**. The positive and negative x-values of the same number will produce the same y-value.

But this is cool because this shows us something else that we can only see when graphing our function. Because the positive x-values and their negative counterpart produce the same result, the graph is symmetric across the y-axis.

Let’s look at a few examples. The first one is for the function **g(x)=x ^{2}-2**.

When we replace the x’s in this function with -x, we end up with this.

This shows us that this function is an even function because when we replace all x values with negative x’s, we end up with the original function.

But let’s take a look at the graph. On the graph, we see that each positive x-value and its negative counterpart result in the same y-value.

The following example is for the function **f(x)=|x|**. This function represents an absolute value function.

When we replace all the x with a negative x, the function does not lose its positive value. We retain the original function.

When we look at the graph of this function, it shows the symmetry across the y-axis.

Now let’s take a look at odd functions.

## What are odd functions?

Odd functions occur when all the x’s in a function are replaced with a negative x, and the results are the negative version of the original function.

**This means that f(-x) = -f(x)**. The positive and negative x-values will produce opposite results.

The graph of an odd function is symmetric across the origin.

Let’s look at a few examples. The first one is for the function **f(x)=x ^{3}**.

When we replace the x with a negative x, the result is the negative version of this function.

This is what that looks like.

When we look at the graph, we can see that the graph is symmetric around the origin. We can see that each positive value results in a y-value, but its negative corresponding x-value results in the negative version.

Let’s look at the function **g(x)=6/x**.

When we replace the x with a negative x, we get **g(-x)=-6/x.**

This answer is the negative version of the original function. Another way of saying this is this function is equal to **g(-x)=-g(x).**

Here is the graph of g(x)=6/x. The function is symmetric with respect to the origin.

Another example is the function **h(x)=1/2(x ^{7})**.

When we replace the x with a negative x, we get **g(-x)=-1/2(x ^{7})**. This represents the negative version of our original function.

The graphed function is symmetric about the origin when we look at it. Here is what that looks like.

## Odd functions, even functions, and neither.

Here are a few things to remember about odd and even functions before moving into the examples.

Make sure you test the function as a whole to know for sure if a function is odd or even. Don’t make assumptions based on what you see.

Also, remember that functions don’t have to be odd or even. We’ll look at an example that cannot be classified as either.

The first example is for the function **f(x)=x ^{3}-3x**.

When all x’s are replaced with negative x’s, we get **f(-x)=-x ^{3}+3x**. This is the negative version of the original function which means that this function is odd.

If we look at the graph, we can see that it is symmetric across the origin.

The following function we’ll look at is **g(x)=x ^{3}+3**.

When we replace all the x’s with negative x’s, we get **g(-x)=-x ^{3}+3**. This is not the original version of the function, but it isn’t the negative version either.

Therefore, this function is neither odd nor even.

The next function is **h(x)= β(16-x ^{2} )**. We replace all the x’s with negative x’s, and we get the original function.

That makes this function an even function. If we look at the graph of this function, it is symmetric around the y-axis.

Now let’s look at the function **f(x)=x-2/x-1**. Most people would assume this function is odd since the exponents are even. But when we replace all x’s with negative x’s, we get something that doesn’t look like the original.

But it doesn’t look like the negative version of the original either. Therefore this function is neither odd nor even.

Let’s look at the next function. We can see that the radical has an odd exponent. When we replace the x with a negative x, we get a -x under the radical. The negative comes out from under the radical because it’s a radical with an odd exponent. This leaves us with the negative version of the original function.

When we look at the graph, we see that the function is symmetric around the origin. This function is odd.

This last example is **f(x)=2**. Constants have the exponent of zero. We can’t change the original version of this function when changing the x to a negative. The function remains the same no matter what we do.

Therefore, this function is even.

Get out there and keep practicing. π

Also, don’t forget to take breaks.