Q: A function $$f is $$one-to-one if and only if:

Different inputs have different outputs: if $$a \neq b$$, then $$f(a) \neq f(b).$$

Question 1

Which of the following statements correctly describes the relationship between a function and its inverse?

Accepted Answer

The graph of a function and its inverse are always reflections of each other across the line $$y = x.$$

Question 2

Given the function $$f(x) = 2x - 3$$, which of the following is the correct formula for its inverse $$f^{-1}(x$$)$$?

Accepted Answer

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

Question 3

Which of the following functions is NOT one-to-one?

Accepted Answer

$$f(x) = x^2$

Question 4

A function $$f is $$one-to-one if and only if:

Accepted Answer

Different inputs have different outputs: if $$a 
eq b$$, then $$f(a) 
eq f(b).$$

Question 5

If $$f(x) = 5x + 8$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x - 8}{5}$$

Question 6

Which test can be used to determine if a function is one-to-one by looking at its graph?

Accepted Answer

Horizontal-line test

Question 7

Suppose $$f(x) = x^2$ and its domain is restricted to x$$ \geq 0$$. What is the inverse function f$$^{-1}$$(x)$$?

Accepted Answer

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

Question 8

Given $$f(x) = 2x + 5$$ and $$g(x) = x - 3$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$(f \circ g)(x) = 2(x - 3) + 5$$

Question 9

If $$f(x) = 3x - 7$$, what is $$(f^{-1}$$ \circ f)(x)$$?

Accepted Answer

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

Question 10

Which of the following is true about the notation $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$)$$ is the inverse function of f(x$$)$$, not the reciprocal.

Question 11

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

Accepted Answer

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

Question 12

If $$f(x) = x^2$ - 2x$$, what is the equation for its inverse relation?

Accepted Answer

$$x = y^2$ - 2y$$

Question 13

Which of the following statements is true about the domain and range of a one-to-one function and its inverse?

Accepted Answer

The domain of $$f is $$the range of $$f^{-1}$$, and the range of $$f is $$the domain of $$f^{-1}.$$

Question 14

A chemical process is modeled by $$f(x) = 4x + 10. If $$the output is 34, what was the input?

Accepted Answer

6

Question 15

Which of the following is a necessary step to find the inverse of a one-to-one function $$f(x)$$?

Accepted Answer

Interchange $$x$$ and $$y in $$the equation.

Question 16

If $$f(x) = x^2$ and f$$^{-1}$$(x) = \sqrt{x}$$, what domain restriction is necessary for $$f^{-1}(x$$)$$ to be a function?

Accepted Answer

$$x \geq 0$$

Question 17

Which of the following compositions always returns the input $$x$$ for a one-to-one function $$f$$ and its inverse $$f^{-1}$$?

Accepted Answer

$$(f \circ f$$^{-1}$$)(x)$$ and $$(f^{-1}$$ \circ f)(x)$$

Question 18

A medication dosage is calculated by $$f(x) = 0.5x + 2$$, where $$x is $$the patient's weight in kg. If a patient receives a dosage of 7 mg, what is their weight?

Accepted Answer

10 kg

Question 19

Which of the following is NOT a property of one-to-one functions?

Accepted Answer

A horizontal line can intersect the graph more than once.

Inverse Functions and One-to-One Functions: Study Notes for College Algebra

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