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Even and Odd Functions

You've probably explored even and odd numbers in the past, but what about even and odd functions? Yes, they exist. No, determining whether a function is even or odd isn't as easy as checking whether the first term is an even or odd number. In this article, we'll see what makes a function even or odd and what that means when it's graphed on the Cartesian plane. Let's get started!

Even and odd functions: the even

A function is considered even if, for each x in the domain f , f - x = f x . For instance:

f - x = x 2

We can test whether this is an even function by subbing -x for x and seeing if we get the same equation.

f - x = - x - x 2 = x 2 = f x

This is indeed an even function. When graphed, even functions have reflective symmetry across the y-axis as illustrated below:

Remember that a function is only even if every potential value of x satisfies the criteria. We cannot find a single value for x where f - x = - f x and call the function even.

Even and odd functions: the odd

A function is considered odd if, for each x in the domain of f, f - x = - f x . Again, we can test it algebraically:

f x = x 3

f - x = - x - x 3 = - x 3 = - f x

f x = x 3 is indeed an odd function. When graphed, odd functions have 180° rotational symmetry around the origin as shown:

The vast majority of functions won't satisfy the criteria for even or odd functions and are therefore neither. Never assume that a function must be even or odd.

Practice questions on even and odd functions

a. Define a function as 3 x 3 + 4 x . Is this function even, odd, or neither?

We need to find f(-x) to determine its symmetry. If f - x = f x , the function is even. If f - x = - f x , the function is odd. Otherwise, the function is neither even nor odd. Let's substitute -x for x:

f - x = 3 - x 3 + 4 - x

We can simplify this using the Power of a Product Property:

f - x = 3 - x 3 + 4 - x

f - x = 3 -1 x 3 + 4 - x

f - x = - 3 x 3 - 4 x

Comparing this with f x = 3 x 3 + 4 x , we find f - x = - 3 x 3 + 4 x = - f x

Since f - x = - f x , the function is odd.

b. Define a function f x = log ( x 2 ) . Is this function even, odd, or neither?

We're working with logarithms this time, but that doesn't change what we have to do. First, we need to determine what f - x is by subbing -x in for x:

f - x = log - - x 2

f - x = log x 2

f - x = f x

Since f - x = f x , this is an even function.

Topics related to the Even and Odd Functions

End Behavior of a Function

Parent Graphs

Degree (of a Polynomial)

Flashcards covering the Even and Odd Functions

Algebra II Flashcards

College Algebra Flashcards

Practice tests covering the Even and Odd Functions

Algebra II Diagnostic Tests

College Algebra Diagnostic Tests

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