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.
$$a)\hspace{.2em}f(x)=2x^3 - 5x - 1$$
$$b)\hspace{.2em}f(x)=\frac{3}{4}x^2 - |x| + 2$$
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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.
$$a)\hspace{.2em}f(x)=\frac{9x}{|x|}$$
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#4:
Instructions: determine if even, odd, or neither.
$$a)\hspace{.2em}f(x)=-3x^{\frac{5}{3}}$$
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#5:
Instructions: determine if even, odd, or neither.
$$a)\hspace{.2em}f(x)=(x - 2)^2 - x$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}neither$$
$$b)\hspace{.2em}even$$
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#2:
Solutions:
$$a)\hspace{.2em}neither$$
$$b)\hspace{.2em}odd$$
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#3:
Solutions:
$$a)\hspace{.2em}odd$$
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#4:
Solutions:
$$a)\hspace{.2em}odd$$
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#5:
Solutions:
$$a)\hspace{.2em}neither$$
