Functions: Domain, Range, Composition and Inverses: Precalculus Math Quiz – PDF Download
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.
🎯 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.
AGraph B only
BGraph A only
CGraph C only
DGraphs 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.
