About Inverse of a Function:
Every one-to-one function has an inverse. To find the inverse of a function, we start by writing f(x) as y. Then we swap the x and y variables. Now, we solve for y. Lastly, we replace y with a special notation: f-1(x), which is read as f inverse of x.
Test Objectives
- Demonstrate the ability to find the inverse of a function
#1:
Instructions: find the inverse of each function.
$$a)\hspace{.2em}f(x)=2x + 1$$
$$b)\hspace{.2em}f(x)=\sqrt[3]{x}- 2$$
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#2:
Instructions: find the inverse of each function.
$$a)\hspace{.2em}f(x)=\frac{3}{7}x - \frac{6}{7}$$
$$b)\hspace{.2em}f(x)=\frac{1}{x - 2}+ 2$$
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#3:
Instructions: find the inverse of each function.
$$a)\hspace{.2em}f(x)=-2 + (x - 1)^3$$
$$b)\hspace{.2em}f(x)=-x^3 - 1$$
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#4:
Instructions: find the inverse of each function.
$$a)\hspace{.2em}f(x)=\sqrt[5]{x}+ 2$$
$$b)\hspace{.2em}f(x)=\frac{2}{x - 2}$$
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#5:
Instructions: find the inverse of each function.
$$a)\hspace{.2em}f(x)=\frac{x - 2}{3x - 1}$$
$$b)\hspace{.2em}f(x)=\frac{x - 1}{2x - 4}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=\frac{x - 1}{2}$$
$$b)\hspace{.2em}f^{-1}(x)=(x + 2)^3$$
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#2:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=\frac{7x + 6}{3}$$
$$b)\hspace{.2em}f^{-1}(x)=\frac{2x - 3}{x - 2}$$
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#3:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=\sqrt[3]{x + 2}+ 1$$
$$b)\hspace{.2em}f^{-1}(x)=\sqrt[3]{-x - 1}$$
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#4:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=(x - 2)^5$$
$$b)\hspace{.2em}f^{-1}(x)=\frac{2x + 2}{x}$$
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#5:
Solutions:
$$a)\hspace{.2em}f^{-1}(x)=-\frac{x - 2}{1 - 3x}$$
$$b)\hspace{.2em}f^{-1}(x)=-\frac{4x - 1}{1 - 2x}$$
