### About Even & Odd Functions:

In some cases, we need to be able to determine if a function is even, odd, or neither. How can we determine if a graph is symmetric with respect to the x-axis? This occurs when we can replace x with -x and obtain the same equation. When a function has this property, f(-x) = f(x), we can say this is an even function. Additionally, we will see that when y can be replaced with -y and we have the same equation, the graph is symmetric with respect to the x-axis. Now a graph that is symmetric with respect to the x-axis, it will not be a function, since it will fail the vertical line test. For a given x-value, it will have a y and -y that is associated. Lastly, we will think about a graph that is symmetric with respect to the origin. For this graph, we can replace x with -x and y with -y and obtain the same equation. A function with this property is known as an odd function. We can show this as: f(-x) = -f(x) or -f(-x) = f(x)

Test Objectives

- Demonstrate the ability to determine if a function is even
- Demonstrate the ability to determine if a function is odd

#1:

Instructions: determine if even, odd, or neither.

$$a)\hspace{.2em}f(x)=2x^3 - 5x - 1$$

$$b)\hspace{.2em}f(x)=\frac{3}{4}x^2 - |x| + 2$$

Watch the Step by Step Video Lesson View the Written Solution

#2:

Instructions: determine if even, odd, or neither.

$$a)\hspace{.2em}f(x)=x \cdot \sqrt[3]{x - 1}$$

$$b)\hspace{.2em}f(x)=\frac{x}{2x^2 - 3}$$

Watch the Step by Step Video Lesson View the Written Solution

#3:

Instructions: determine if even, odd, or neither.

$$a)\hspace{.2em}f(x)=\frac{9x}{|x|}$$

Watch the Step by Step Video Lesson View the Written Solution

#4:

Instructions: determine if even, odd, or neither.

$$a)\hspace{.2em}f(x)=-3x^{\frac{5}{3}}$$

Watch the Step by Step Video Lesson View the Written Solution

#5:

Instructions: determine if even, odd, or neither.

$$a)\hspace{.2em}f(x)=(x - 2)^2 - x$$

Watch the Step by Step Video Lesson View the Written Solution

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}neither$$

$$b)\hspace{.2em}even$$

Watch the Step by Step Video Lesson

#2:

Solutions:

$$a)\hspace{.2em}neither$$

$$b)\hspace{.2em}odd$$

Watch the Step by Step Video Lesson

#3:

Solutions:

$$a)\hspace{.2em}odd$$

Watch the Step by Step Video Lesson

#4:

Solutions:

$$a)\hspace{.2em}odd$$

Watch the Step by Step Video Lesson

#5:

Solutions:

$$a)\hspace{.2em}neither$$