# Even and Odd Functions Worksheets

There are three common classifications of functions: even, odd, and neither. You can determine their nature by either using algebra or graphing them. If you graph them even functions are symmetric with respect to the y-axis. This means that if you pick two opposite input values (like -4 and 4), the function value will stay the same along the curve. When graphing odd functions, they will stay symmetric with respect to the origin. Algebraically we can put a function to the stress test by substituting -x values in for f(x). If the opposite input results in the same output, it is classified as even. If the opposite input gives the opposite output, it is classified as odd. Students can use these worksheets and lesson to learn how to identify a function as even or odd.

### Aligned Standard: HSF-IF.C.8

- Evens or Odds Step-by-step Lesson- You are given a straight polynomial and asked to classify it.
- Guided Lesson - Graphing these helps out a ton! See if it helps you. Classify the functions as even, odd, or neither.
- Guided Lesson Explanation - We are just looking for a symmetrical or asymmetrical tendency with this one.
- Practice Worksheet - Read the functions and classify them as even, odd, or neither.
- Matching Worksheet - You can draw a line, if you would like. See how quickly you can do this.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Classify the functions as even, odd, or neither.

- Homework 1 - Classify the functions as even, odd, or neither: f (x) = -3x
^{2}+ 4. - Homework 2 - It all starts after you graph the function.
- Homework 3 - This graphing example makes a nice "U" shape.

### Practice Worksheets

Graphing them is a must!

- Practice 1 - If the graph is symmetric along the y-axis, it is classified as even.
- Practice 2 - If it is symmetric along origin (0, 0) it is classified as odd.
- Practice 3 - If the function does not meet either measure asymmetric to these tendencies, it is classified as neither.

### Math Skill Quizzes

You should be able to recognize some trends in the graphs as you move forward.

- Quiz 1 - Is f (x)=x
^{4}+ x^{9}+ x : even, odd, or neither? - Quiz 2 - These mostly are symmetric about the y-axis and all the exponents are even, that makes them classified as even.
- Quiz 3 - I think that it is funny that symmetric graphs are odd. Kind of defies logic?

### How Do You Classify Functions?

In math a function is an expression or rule set that defines a distinct relationship between an input an output variable. They are common stated it equation form as f(x). Just like the concept of even and odd numbers, functions have an even or odd nature to them as well.

By graphical definition, a function is even if the graph is symmetric with respect to the y-axis, while, algebraically, a function is even if f(-x) = f(x) for all the x present in the domain. You can evaluate if a function is even by substituting negative x in for f(x). If you get the starting function, that assures you that the function is even. While a function is odd if the function's graph is symmetric to the origin. While algebraically, the function is only odd if f(-x) = -f(x) for all the x present in the domain. To put that equation words, if you substitute negative x for f(x) and get the opposite (negative value) of the starting function it is odd. If we evaluate a function by substituting negative x and we do not receive what we started with or the opposite, it would be classified as neither odd nor even.

Let us consider a function with two values of x and attempt to classify it.

f(x) = x^{3} + 4x + 2 , if x = -x, then the equation will become:

f(-x) = (-x)^{3} + 4(-x) + 2, f(-x) = (-x)^{3} - 4x + 2

The value of the function changed, which means that this is an odd function.

f(x) = x^{2}+2, if x = - x, then the equation will become:

f(-x) = (-x)^{2} + 2, f(-x) = x^{2} + 2, f(-x) = f(x).

The value of the function does not change, which means that this is an even function.

### Why is Classifying Functions Important at All?

As we progress forward with precalculus it is often important to understand the connection between algebra and geometry. This reminds us how the two seemingly different areas relate and how as we advance, we can transverse between these two worlds. In this case we can visualize algebraic statements and geometric statements in the form of a graph. As students start to make their way into the world of calculus, they will be able to use this skill to reduce complex integrals to zero by understanding this simple classification skill. It is something you want to know for those rainy days in calculus class.