Question 1

Which of the following statements best describes a one-to-one function?

Accepted Answer

A function where every output corresponds to exactly one input, and different inputs always have different outputs.

Question 2

Given the function $$f(x) = x^6$, use the horizontal line test to determine if f$ is one-to-one.

Accepted Answer

No, because some horizontal lines intersect the graph more than once.

Question 3

Suppose $$f(x) = 3x - 5. $$What is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{1}{3}(x + 5)$$

Question 4

If a function $$f is $$increasing on an interval $$I$$, what can you conclude about $$f on$$ $$I$$?

Accepted Answer

$$f is $$one-to-one on $$I.$$

Question 5

Given the mapping: California $$\rightarrow 39.3$$, Texas $$\rightarrow 27.9$$, Florida $$\rightarrow 20.6$$, New York $$\rightarrow 19.7$$, Pennsylvania $$\rightarrow 12.8$$, what is the domain of the inverse function?

Accepted Answer

The set of populations: 39.3, 27.9, 20.6, 19.7, 12.8

Question 6

Which of the following is true about the graphs of a one-to-one function $$f$$ and its inverse $$f^{-1}$$?

Accepted Answer

They are symmetric with respect to the line $$y = x.$$

Question 7

Given $$f(x) = x^2$ + 1$$ with domain $$x \geq 0$$, find the explicit form of the inverse function $$f^{-1}(x$$)$$.

Accepted Answer

$$f^{-1}(x$$) = \sqrt{x - 1}$$

Question 8

If $$f$$ and $$g$$ are inverse functions, which of the following must be true for all $$x in $$the domain of $$f$$?

Accepted Answer

$$f(g(x)) = x$$ and $$g(f(x)) = x$$

Question 9

A hospital uses a function $$f to $$convert Celsius temperatures to Fahrenheit: $$f(x) = \frac{9}{5}x + 32. $$What is the inverse function $$f^{-1}(x$$)$$ to convert Fahrenheit to Celsius?

Accepted Answer

$$f^{-1}(x$$) = \frac{5}{9}(x - 32)$$

Question 10

Suppose $$f(x) = \frac{2x + 3}{x - 1} is $$one-to-one. What is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

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

Question 11

A function $$f is $$defined by the set of ordered pairs $$\{(1, 4), (2, 5), (3, 6)\}. $$What is the inverse function $$f^{-1}$$?

Accepted Answer

$$\{(4, 1), (5, 2), (6, 3)\}$$

Question 12

Which of the following is a necessary condition for a function to have an inverse?

Accepted Answer

The function must be one-to-one.

Question 13

If $$f(x) is a $$one-to-one function with domain $$D$$ and range $$R$$, what are the domain and range of $$f^{-1}(x$$)$$?

Accepted Answer

Domain: $$R$$, Range: $$D$$

Question 14

A pharmaceutical company models the concentration of a drug in the bloodstream by $$C(t) = 5e$$^{-0.2t}$$, where t$ is time in hours. If the safe concentration is 1 mg/L, how long after administration will the concentration reach this level? (Solve for t$.)

Accepted Answer

$$t = \frac{\ln(5)}{0.2}$$

Question 15

Which of the following best describes the process for finding the inverse of a one-to-one function $$f$$ defined by an equation?

Accepted Answer

Interchange $$x$$ and $$y in $$the equation, then solve for $$y in $$terms of $$x.$$

Question 16

If $$f(x) is $$not one-to-one on its entire domain, but is one-to-one on a restricted domain, what can be said about the existence of $$f^{-1}(x$$)$$?

Accepted Answer

$$f$$ has an inverse function only on the restricted domain.

One-to-One and Inverse Functions: Precalculus Study Notes

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