Precalculus Functions Review Quiz: 10 Questions

Contents

Key takeaway: This precalculus functions review quiz poses 10 multiple-choice questions on domain, range, composition and inverses. Several items use graphs or mapping diagrams, so read the labels first. The quiz builds on F-IF.A.1 and F-BF.B.4. Explanations show why each wrong choice tempts students. Give yourself about 15 minutes.

Answer these 10 questions on domain, composition, decomposition, inverse functions and restricted domains. Each question has exactly one correct answer, and the explanation names the most common mistake. Aim for at least 8 correct.

PrecalculusCommon Core: F-IF.A.1 · F-BF.A.1c · F-BF.B.4b · F-BF.B.4c · F-BF.B.4d

🎯 Functions: Domain, Range, Composition and Inverses – Precalculus math quiz

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Question 1

What is the domain of \(f(x)=\frac{\sqrt{x-3}}{x-7}\)?

A \([3,\infty)\)

B \([3,7)\) ∪ \((7,\infty)\)

C \((3,7)\) ∪ \((7,\infty)\)

D \((-\infty,7)\) ∪ \((7,\infty)\)

Question 2

The graph shows a function \(f\) that starts at \((1,2)\) and heads down and to the right. What is its range?

Graph of the function.

A \([2,\infty)\)

B \([1,\infty)\)

C \((-\infty,2]\)

D \((-\infty,1]\)

Question 3

Let \(f(x)=x^{2}-1\) and \(g(x)=x+3\). Which expression equals \(f(g(x))\)?

A \(x^{2}+2\)

B \(x^{2}+6x+8\)

C \(x^{2}+8\)

D \(x^{2}+6x+10\)

Question 4

In the mapping diagram, \(g\) sends 1, 2, 3 to 2, 5, 8 and \(f\) sends 2, 5, 8 to 3, 24, 63. What is \(f(g(2))\)?

Mapping diagram for g (set A to set B) followed by f (set B to set C).

A \(24\)

B \(5\)

C \(2\)

D \(63\)

Question 5

Which pair of functions gives \(h(x)=\frac{1}{(x-3)^{2}}\) when \(h(x)=f(g(x))\)?

A \(f(x)=\frac{1}{x-3}\) and \(g(x)=x^{2}\)

B \(f(x)=\frac{1}{x}\) and \(g(x)=x^{2}-3\)

C \(f(x)=\frac{1}{x^{2}}\) and \(g(x)=x-3\)

D \(f(x)=x-3\) and \(g(x)=\frac{1}{x^{2}}\)

Question 6

What is the inverse function of \(f(x)=\frac{x+5}{2}\)?

A \(f^{-1}(x)=\frac{2}{x+5}\)

B \(f^{-1}(x)=2x+5\)

C \(f^{-1}(x)=\frac{x-5}{2}\)

D \(f^{-1}(x)=2x-5\)

Question 7

The graph of a one-to-one function \(f\) passes through \((-3,-2)\), \((-1,0)\), \((1,2)\), \((3,3)\) and \((5,6)\). What is \(f^{-1}(2)\)?

Graph of f.

A \(2\)

B \(3\)

C \(\frac{1}{2}\)

D \(1\)

Question 8

Which graph has an inverse function?

Three graphs labeled A, B and C.

A Graph B only

B Graph A only

C Graph C only

D Graphs A and B

Question 9

What is the inverse of \(f(x)=(x-4)^{2}\) for \(x \ge 4\)?

A \(f^{-1}(x)=4-\sqrt{x}\)

B \(f^{-1}(x)=\sqrt{x}-4\)

C \(f^{-1}(x)=4+\sqrt{x}\)

D \(f^{-1}(x)=\sqrt{x+4}\)

Question 10

If \(f(4)=-1\) for a one-to-one function \(f\), which statement must be true?

A \(f^{-1}(-1)=4\)

B \(f^{-1}(4)=-1\)

C \(f^{-1}(-1)=\frac{1}{4}\)

D \(f^{-1}(1)=-4\)

🎉 Quiz complete!

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Remember:

Standards F-IF.A.1, F-BF.A.1c, F-BF.B.4b, F-BF.B.4c and F-BF.B.4d: notation, composition, inverses and restricted domains.
Domain rules: no zero denominators, no negative numbers under even roots, and only positive inputs for \(\ln\).
In \(f(g(x))\), the input must be in the domain of \(g\) and \(g(x)\) must be in the domain of \(f\).
\(f^{-1}(f(x))=x\) and \(f(f^{-1}(x))=x\); the domain of \(f\) is the range of \(f^{-1}\) and the reverse.
A function has an inverse function only if it passes the horizontal line test; restrict the domain when it fails.

Which questions cover domain and range?

The first question asks for the domain of \(\frac{\sqrt{x-3}}{x-7}\), so combine the root condition with the excluded 7 to get \([3,7)\cup(7,\infty)\). The graph in question 2 starts at \((1,2)\) and falls, which gives a range of \((-\infty,2]\). Remember that range reads heights, not widths.

How do the composition questions work?

With \(f(x)=x^{2}-1\) and \(g(x)=x+3\), question 3 gives \(x^{2}+6x+8\). Many students square only the \(x\) and miss the middle term. A mapping diagram in question 4 sends 2 to 5 and then 5 to 24, so the answer is 24. Question 5 decomposes \(\frac{1}{(x-3)^{2}}\) into \(\frac{1}{x^{2}}\) and \(x-3\).

What do the inverse questions need?

Question 6 inverts \(\frac{x+5}{2}\) to get \(2x-5\). Question 8 applies the horizontal line test to three graphs, and only Graph B qualifies. In question 9, the restricted parabola \((x-4)^{2}\) has inverse \(4+\sqrt{x}\). Finally, question 10 states the key swap: if \(f(4)=-1\), then \(f^{-1}(-1)=4\).

What should you do with your result?

Strong results suggest the unit paper is within reach. After several misses, note whether they were about domain or inverses, then reread that lesson section. Retake the quiz in a day and compare.