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.