Precalculus Inverse Functions Test Solutions and Scoring (PDF)
This inverse functions test answers for Precalculus covers inverse functions (Common Core F-BF.B.4a, F-BF.B.4b, F-BF.B.4c, F-BF.B.4d), with a free PDF to print.
Precalculus · Common Core State Standards: F-BF.B.4a, F-BF.B.4b, F-BF.B.4c, F-BF.B.4d
Inverse functions test answers: worked solutions
Question 1 – Horizontal line test [2 pts]
The horizontal line \(y=4\) meets the graph at \((-2,4)\) and \((2,4)\).
No, it is not one-to-one
Scoring: 1 point for the answer, 1 point for the horizontal line reason.
Question 2 – Inverse of a line [2 pts]
Swap: \(x=\frac{2y-5}{3}\), so \(3x=2y-5\) and \(y=\frac{3x+5}{2}\).
\(f^{-1}(x)=\frac{3x+5}{2}\)
Scoring: 1 point for swapping, 1 point for solving.
Question 3 – Inverse of a root function [2 pts]
\(x=\sqrt{y+3}\) gives \(y=x^{2}-3\), with \(x\ge 0\).
\(f^{-1}(x)=x^{2}-3,\ x\ge 0\)
Scoring: 1 point for the formula, 1 point for the domain.
Question 4 – Verify inverses [2 pts]
\(f(g(x))=5\cdot\frac{x+2}{5}-2=x\). \(g(f(x))=\frac{5x-2+2}{5}=x\).
Both equal x
Scoring: 1 point for each composition.
Question 5 – Points on an inverse [3 pts]
(a) Swap each pair: \((1,0)\), \((2,2)\), \((3,4)\). (b) \(x=\frac{y}{2}+1\) gives \(y=2x-2\). (c) They are reflections over the line \(y=x\).
\((1,0),(2,2),(3,4)\); \(f^{-1}(x)=2x-2\); reflection over \(y=x\)
Scoring: 1 point each for (a), (b) and (c).
Question 6 – Rational inverse [2 pts]
Swap: \(x(y-2)=y+1\), so \(xy-2x=y+1\), \(y(x-1)=2x+1\).
\(f^{-1}(x)=\frac{2x+1}{x-1}\)
Scoring: 1 point for setting up and clearing the fraction, 1 point for the result.
Question 7 – Restricted domain [2 pts]
Swap: \((y+1)^{2}=x+4\). Since \(y+1\ge 0\), \(y=\sqrt{x+4}-1\).
\(f^{-1}(x)=\sqrt{x+4}-1\)
Scoring: 1 point for isolating the square, 1 point for the correct root sign.
Question 8 – Car value model [3 pts]
(a) \(t=\frac{28000-V}{2000}\). (b) \(\frac{11000}{2000}=5.5\). (c) \(V^{-1}\) gives the age of the car for a given value.
\(V^{-1}(V)=\frac{28000-V}{2000}\); \(5.5\) years; age from value
Scoring: 1 point for each part.
Question 9 – Domain and range of an inverse [2 pts]
\(x-2=\sqrt{y-5}\), so \(y=(x-2)^{2}+5\). Outputs of \(f\) are at least \(2\), so the domain of \(f^{-1}\) is \(x\ge 2\).
\(f^{-1}(x)=(x-2)^{2}+5,\ x\ge 2\)
Scoring: 1 point for the formula, 1 point for the domain.
Total score
20 points
Key facts behind this inverse functions test answers
- f^-1(f(x)) = x and f(f^-1(x)) = x.
- Find an inverse: write y = f(x), swap x and y, solve for y.
- Domain of f becomes the range of f^-1, and the range of f becomes the domain of f^-1.
- The graph of f^-1 is the reflection of the graph of f over y = x.
- Restrict the domain of a function that is not one-to-one, such as x^2 with x >= 0.
How to use this inverse functions test answers
Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.
All Precalculus chapters: Precalculus math.
