### 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 y-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. A graph that is symmetric with respect to the x-axis, does not pass the vertical line test, and will not be a function. 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.

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

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

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#2:

Instructions: determine if even, odd, or neither.

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

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

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#3:

Instructions: determine if even, odd, or neither.

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

$$b)\hspace{.2em}f(x)=x^4\sqrt{1 - x^2}$$

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#4:

Instructions: determine if even, odd, or neither.

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

$$b)\hspace{.2em}f(x)=\frac{1}{3}x^6 - 3x^2$$

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#5:

Instructions: determine if even, odd, or neither.

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

$$b)\hspace{.2em}f(x)=x + \frac{1}{x}$$

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Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}\text{neither}$$ Desmos Link for More Detail

$$b)\hspace{.2em}\text{even}$$ Desmos Link for More Detail

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#2:

Solutions:

$$a)\hspace{.2em}\text{neither}$$ Desmos Link for More Detail

$$b)\hspace{.2em}\text{odd}$$ Desmos Link for More Detail

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#3:

Solutions:

$$a)\hspace{.2em}\text{odd}$$ Desmos Link for More Detail

$$b)\hspace{.2em}\text{even}$$ Desmos Link for More Detail

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#4:

Solutions:

$$a)\hspace{.2em}\text{odd}$$ Desmos Link for More Detail

$$b)\hspace{.2em}\text{even}$$ Desmos Link for More Detail

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#5:

Solutions:

$$a)\hspace{.2em}\text{neither}$$ Desmos Link for More Detail

$$b)\hspace{.2em}\text{odd}$$ Desmos Link for More Detail