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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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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!

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.

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.

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*.

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.

Take, for example, this equation *y* = *x*^2. Can you identify if this equation is even or odd? First, replace the *x* value with *-x*. Now evaluate the equation. Since *x* squared means 'x times x' and a negative times a negative is a positive, we will have the same equation as the original. Therefore, our equation is an even function.

Take a look at the graph of this equation, and you can see it matches our definition of an even function.

Okay, how about this equation, *y* = *x*^3? Can you identify if this equation is even or odd? Replace the *x* value with *-x* and the *y* with *-y*. Now evaluate the equation. *x* cubed is the same thing as *x* times *x* times *x*. Which would give us a negative times a negative, which is a positive times a negative.

That would give us a *-x* value. We can't do anything else to the *-y* value. Therefore, the equation is left with a *-x* value and a *-y* value, the exact opposite of the original equation. The equation represents an odd function.

Here is the graph of this equation. You can see that when the graph is rotated, it matches the definition of an odd function.

If you insert a -*x* into an equation and you don't get the original equation, and you insert a -*x* and a -*y* and not all of the values have been changed from negative to positive or positive to negative, then the equation is neither even nor odd.

A quick trick for even and odd functions is to analyze the exponents in the equation. If the exponents for the *x* values in the equation equal an even number, then the function is even. If the exponents for the *x* values and the *y* values in the equation equal an odd number, then the function is odd. This, however, does not apply to absolute value equations.

If you had an equation that was *y* = |*x*^3| and you replaced the *x* value with a *-x* value, you would still have a positive *x*. That is because every time you evaluate an equation in absolute value, you will always have a positive result. It is better to test the equation than to depend on a trick, just in case.

In conclusion, you can identify symmetrical equations by using even and odd functions. Even functions are functions that are symmetrical across the *y*-axis. You can identify even functions by replacing the *x* value in the equation with a negative *x* value. If you evaluate the equation and get the same as the original, then the equation is even.

Odd functions are functions that are symmetrical across the origin. You can identify odd functions by replacing the *x* and the *y* values in the equation with *-x* and *-y* values. If you evaluate the equation and get the exact opposite equation as the original (positives are negatives and negatives are positives), then the equation is odd.

You can use exponents as a trick to identify even and odd functions. Often, even functions have even exponents and odd functions have odd exponents, but there are exceptions to this rule, so it is always better to double check!

By completing this lesson, you might become better-prepared to:

- Compare even and odd functions
- Solve symmetrical equations
- Determine whether a function is odd or even
- Use exponents to identify even and odd functions

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

- Recognizing Symmetry Graphically, Algebraically, and Numerically About the X-axis and Y-axis 11:19
- Recognizing Symmetry Graphically, Algebraically & Numerically About the Origin 6:17
- Identifying the Line of Symmetry: Definition & Examples 8:07
- Even & Odd Functions: Definition & Examples 10:02
- Go to Graph Symmetry

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