### About Inverse Functions:

In this section, we test our ability to determine if a function is one-to-one. A function is one-to-one if each y-value of the function corresponds to only one x-value. We can use the horizontal line test to determine if any y-value corresponds to more than one x-value. When a function is one-to-one, we can find the inverse by interchanging the x and y values.

Test Objectives

- Demonstrate the ability to sketch the graph of a function
- Demonstrate the ability to determine if a function is one-to-one
- Demonstrate the ability to find the inverse of a function

#1:

Instructions: Determine if the function is one-to-one.

a) $$f(x) = x^3 + 2$$

b) $$f(x) = 2x^2 - 1$$

c) $$h(x) = 3|x| - 2$$

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#2:

Instructions: Determine if the function is one-to-one.

a) $$f(x) = -x^2 - 3$$

b) $$f(x) = \sqrt{x} + 1$$

c) $$f(x) = -4x + 5$$

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#3:

Instructions: Find the inverse.

a) $$h(x) = \frac{- 3}{x - 2} + 2$$

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#4:

Instructions: Find the inverse.

a) $$f(x) = \frac{1}{x + 3} - 1$$

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#5:

Instructions: Determine if the functions are inverses.

a) $$f(x) = x - 1$$ $$h(x) = x + 1$$

b) $$g(x) = \frac{2}{x} + 3$$ $$f(x) = \frac{2}{x - 3}$$

c) $$f(x) = 1 - \frac{1}{3}x$$ $$g(x) = \frac{-2x + 2}{3}$$

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Written Solutions:

#1:

Solutions:

a) Yes, this function is one-to-one.

b) No, this function is not one-to-one.

c) No, this function is not one-to-one.

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#2:

Solutions:

a) No, this function is not one-to-one.

b) Yes, this function is one-to-one.

c) Yes, this function is one-to-one.

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#3:

Solutions:

a) $$h^{-1}(x) = \frac{2x - 7}{x - 2}$$

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#4:

Solutions:

a) $$f^{-1}(x) = \frac{-3x - 2}{x + 1}$$

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#5:

Solutions:

a) f(x) and h(x) are inverses.

b) g(x) and f(x) are inverses.

b) f(x) and g(x) are not inverses.