# Even & Odd Functions: Definition & Examples Video

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• 0:01 Why Use Even and Odd…
• 2:47 Defining Even and Odd…
• 3:17 Identifying Functions…
• 4:39 Identifying Functions…
• 8:46 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Even and odd functions are an aspect of line symmetry. You can use knowledge of even and odd functions to quickly identify equation symmetry. After this lesson, test your knowledge with a short quiz!

## Why Use Even and Odd Functions?

Brandon is trying to solve some algebra problems quickly. His teacher has asked him to identify the problems that are symmetrical across the y-axis and the problems that are symmetrical across the origin. He can use what he knows about even and odd functions to identify symmetry in his math homework.

Even and odd functions can help you quickly identify the graphs of functions and vice versa. A function is an equation that shows a unique relationship between the x and the y values.

A function cannot have the same x value for a y value. For example, if you were shipping several packages that were the same size and weight, you would probably pay more money for the distance the package has to travel. In this case, the x value would be the distance the package has to travel, and the y value would be the cost of the package. Since the package can only travel one distance at a time and can't be in two places at once, this would be an example of a function. You would only pay one price for one package to go a certain distance, no more.

In this case, the cost would be the y-axis and the distance would be the x-axis. The further the package would have to travel, the more you would have to pay. You have a unique x value for a unique y value. You wouldn't have to pay two different prices for the same distance traveled.

However, you can have two y values that are the same in a function. Let's say that your house sat on the origin, and you needed a package to travel twenty miles west of your house and twenty miles east of your house. In this case, even though the packages are traveling in different directions, they are going the same distance. Therefore, the cost of the shipping would be the same.

You can perform what is called the vertical line test to determine if a graph is a function. If you draw a vertical line anywhere on the graph, and the vertical line hits more than one point, then the line is not a function. Again, the vertical line can be drawn anywhere on the graph. Just because it doesn't hit more than one point on one side of the graph, doesn't mean that the line is a function.

## Defining Even and Odd Functions

So, let's get back to even and odd functions.

An even function is when the left half of the graph exactly reflects the line or shape on the right half of the graph, as shown in this graph.

An odd function is when the left half of the graph exactly reflects the line or shape on the right half of the graph, except it is upside down, as shown in this graph.

## Identifying Functions Graphically

Even functions are always symmetrical about the y-axis. Take a look at this graph; does this graph represent an even or odd function?

If we look back at the definition of an even function, it tells us that the left half of the graph should match the right half. In this case, both sides of the graph match one another. Therefore, this graph represents an even function. The equation of this graph is y = |x|.

Odd functions are always symmetrical about the origin. Take a look at this graph; does this graph represent an even or odd function?

If we look back at the definition of an odd function, it tells us that the left half of the graph should match the right half, except upside down. Let's take this graph, and rotate it upside down. We can see that the two halves of the graph match exactly. Therefore, this graph represents an odd function. The equation of this graph is y = 1/x.

## Identifying Functions Algebraically

You can also identify even and odd functions by analyzing function equations. This is helpful when you don't have the graph of the equation to evaluate or you have to identify equations quickly. An even function can be identified by replacing the x value in the function with a -x value. If you evaluate the equation and end up with the original equation, then the function is an even function.

An odd function can be identified by replacing both the x and y values with -x and -y values. If the values in your equation are the opposite (positives are negatives and negatives are positives), then the function is odd.

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