Precalculus Inverse Functions Worksheet Solutions (PDF)
This inverse functions 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 answers: exercises 1 to 10
Exercise 1 – One-to-one table
The output \(6\) comes from both \(x=1\) and \(x=3\).
No, it is not one-to-one
Exercise 2 – Inverse of a line
\(y=3x-6\). Swap: \(x=3y-6\), so \(y=\frac{x+6}{3}\).
\(f^{-1}(x)=\frac{x+6}{3}=\frac{x}{3}+2\)
Exercise 3 – Inverse of a quotient by a constant
Swap: \(x=\frac{y+4}{5}\), so \(5x=y+4\).
\(f^{-1}(x)=5x-4\)
Exercise 4 – Inverse of a cubic
Swap: \(x=y^{3}+1\), so \(y^{3}=x-1\).
\(f^{-1}(x)=\sqrt[3]{x-1}\)
Exercise 5 – Inverse of a rational function
Swap: \(x=\frac{2}{y-1}\), so \(y-1=\frac{2}{x}\).
\(f^{-1}(x)=\frac{2}{x}+1\)
Exercise 6 – Inverse of a root
Swap: \(x=\sqrt{y-2}\). Square: \(y=x^{2}+2\). Outputs of \(f\) are at least \(0\), so the new domain is \(x\ge 0\).
\(f^{-1}(x)=x^{2}+2,\ x\ge 0\)
Exercise 7 – Inverse of a restricted parabola
Swap: \(x=(y-3)^{2}\). Since \(y\ge 3\), \(y-3=\sqrt{x}\).
\(f^{-1}(x)=\sqrt{x}+3,\ x\ge 0\)
Exercise 8 – Verify inverses
\(f(g(x))=4\cdot\frac{x-7}{4}+7=x\). \(g(f(x))=\frac{4x+7-7}{4}=x\).
Both compositions equal x
Exercise 9 – Read a table of an inverse
Find the output in the second row: \(11\) pairs with \(x=3\), and \(5\) with \(x=1\). \(f^{-1}(14)=4\), so \(f(4)=14\).
\(3,\ 1,\ 14\)
Exercise 10 – Equation from a graph
The orange line is the inverse. Solve \(x=2y+1\) for \(y\).
\(y=\frac{x-1}{2}\)
Exercises 11 and up
Exercise 11 – Domain and range of an inverse
\(f\): domain \([-1,\infty)\), range \([0,\infty)\). Swap: \(x=\sqrt{y+1}\), so \(y=x^{2}-1\). The domain and range swap.
\(f^{-1}(x)=x^{2}-1\) with domain \([0,\infty)\) and range \([-1,\infty)\)
Exercise 12 – Restrict the domain
Swap: \(x=y^{2}-4\), so \(y^{2}=x+4\) and \(y=\sqrt{x+4}\) because \(y\ge 0\).
\(f^{-1}(x)=\sqrt{x+4},\ x\ge -4\)
Exercise 13 – Temperature conversion
\(F-32=1.8C\), so \(C=\frac{F-32}{1.8}\). For \(F=95\): \(\frac{63}{1.8}=35\).
\(C=\frac{F-32}{1.8}\); \(35^{\circ}\text{C}\)
Exercise 14 – Rational inverse and check
Swap: \(x=\frac{y+2}{y-3}\), so \(xy-3x=y+2\) and \(y(x-1)=3x+2\). Then \(y=\frac{3x+2}{x-1}\). Check: \(f(5)=\frac{7}{2}\) and \(f^{-1}\left(\frac72\right)=\frac{10.5+2}{2.5}=5\).
\(f^{-1}(x)=\frac{3x+2}{x-1}\)
Exercise 15 – Complete the square first
\(y=(x+3)^{2}-9\). Swap: \(x=(y+3)^{2}-9\), so \((y+3)^{2}=x+9\). Since \(y+3\ge 0\): \(y=\sqrt{x+9}-3\).
\(f^{-1}(x)=\sqrt{x+9}-3,\ x\ge -9\)
Exercise 16 – Another rational inverse
Swap: \(x=\frac{y}{y+2}\), so \(xy+2x=y\), \(y(x-1)=-2x\), \(y=\frac{-2x}{x-1}\).
\(f^{-1}(x)=\frac{2x}{1-x}\)
Exercise 17 – Taxi fare inverse
\(C-4=2.5m\), so \(m=\frac{C-4}{2.5}\). It gives miles for a fare. For \(C=29\): \(\frac{25}{2.5}=10\).
\(m=\frac{C-4}{2.5}\); \(10\) miles
Exercise 18 – Car value
\(V=24000-1800t\), so \(t=\frac{24000-V}{1800}\). For \(V=15000\): \(t=\frac{9000}{1800}=5\). For \(V=12000\): \(t=\frac{12000}{1800}=\frac{20}{3}\approx 6.67\).
\(t=\frac{24000-V}{1800}\); \(5\) years; about \(6.67\) years
Exercise 19 – Find f from its inverse
Swap and solve: \(x=\frac{y-3}{2}\) gives \(y=2x+3\). Check: \(f(1)=5\) and \(f^{-1}(5)=\frac{2}{2}=1\).
\(f(x)=2x+3\)
Exercise 20 – Where f meets its inverse
Swap coordinates: \((4,2)\). Solve \(\frac{x}{2}+3=x\): \(3=\frac{x}{2}\), so \(x=6\).
\((4,2)\); \((6,6)\)
Key facts behind this inverse functions 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 answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
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Download the inverse functions answers as a PDF with the button below and keep it with your notes.
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