Lesson Objectives

- Learn how to determine if a function is even
- Learn how to determine if a function is odd

## How to Determine if a Function is Even, Odd, or Neither

### Even Functions

A function is even if it is symmetric with respect to the y-axis. $$f(x)=f(-x)$$ Let's look at a few examples.

Example #1: Determine if the function is even. $$f(x)=-5x^2 - 3x + 1$$ Let's plug in a -x in for x and see if f(-x) = f(x). $$f(-x)=-5(-x)^2 - 3(-x) + 1$$ $$f(-x)=-5x^2 + 3x + 1$$ As we can see, f(-x) is not equal to f(x): $$f(x)=-5x^2 - 3x + 1$$ $$f(-x)=-5x^2 + 3x + 1$$ Therefore, we can conclude this function is not even.

Example #2: Determine if the function is even. $$f(x)=2x^4 + x^2 - 1$$ Let's plug in a -x in for x and see if f(-x) = f(x). $$f(-x)=2(-x)^4 + (-x)^2 - 1$$ $$f(-x)=2x^4 + x^2 - 1$$ As we can see, f(x) is equal to f(-x): $$f(x)=2x^4 + x^2 - 1$$ $$f(-x)=2x^4 + x^2 - 1$$ Therefore, we can conclude this function is even.

Example #3: Determine if the function is odd. $$f(x)=x^3 - 7x$$ Let's plug in and see if -f(x) = f(-x). $$f(-x)=(-x)^3 - 7(-x)$$ $$f(-x)=-x^3 + 7x$$ Now, let's find -f(x) and see if they match. $$-f(x)=-1 \cdot (x^3 - 7x)$$ $$-f(x)=-x^3 + 7x$$ As we can see, -f(x) is equal to f(-x). $$f(-x)=-x^3 + 7x$$ $$-f(x)=-x^3 + 7x$$ Therefore, we can conclude this function is odd.

Example #1: Determine if the function is even. $$f(x)=-5x^2 - 3x + 1$$ Let's plug in a -x in for x and see if f(-x) = f(x). $$f(-x)=-5(-x)^2 - 3(-x) + 1$$ $$f(-x)=-5x^2 + 3x + 1$$ As we can see, f(-x) is not equal to f(x): $$f(x)=-5x^2 - 3x + 1$$ $$f(-x)=-5x^2 + 3x + 1$$ Therefore, we can conclude this function is not even.

Example #2: Determine if the function is even. $$f(x)=2x^4 + x^2 - 1$$ Let's plug in a -x in for x and see if f(-x) = f(x). $$f(-x)=2(-x)^4 + (-x)^2 - 1$$ $$f(-x)=2x^4 + x^2 - 1$$ As we can see, f(x) is equal to f(-x): $$f(x)=2x^4 + x^2 - 1$$ $$f(-x)=2x^4 + x^2 - 1$$ Therefore, we can conclude this function is even.

### Odd Functions

A function is odd if it is symmetric with respect to the origin. $$f(-x)=-f(x)$$ Let's look at an example.Example #3: Determine if the function is odd. $$f(x)=x^3 - 7x$$ Let's plug in and see if -f(x) = f(-x). $$f(-x)=(-x)^3 - 7(-x)$$ $$f(-x)=-x^3 + 7x$$ Now, let's find -f(x) and see if they match. $$-f(x)=-1 \cdot (x^3 - 7x)$$ $$-f(x)=-x^3 + 7x$$ As we can see, -f(x) is equal to f(-x). $$f(-x)=-x^3 + 7x$$ $$-f(x)=-x^3 + 7x$$ Therefore, we can conclude this function is odd.

#### Skills Check:

Example #1

Determine if even, odd, or neither. $$f(x)=-4x^3 - \frac{1}{2}x + 2$$

Please choose the best answer.

A

Even

B

Odd

C

Neither

Example #2

Determine if even, odd, or neither. $$f(x)=5|x| + x^2$$

Please choose the best answer.

A

Even

B

Odd

C

Neither

Example #3

Determine if even, odd, or neither. $$f(x)=-x^3 - 3\sqrt[3]{x}$$

Please choose the best answer.

A

Even

B

Odd

C

Neither

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