Find the matching inverse input
A depth read backward
The curve shows the depth of water in a tank. The horizontal axis is the time in minutes, and the vertical axis is the depth in centimeters.
Drag the orange point along the curve. At 2 minutes the point is at height 10, so the depth is 10 centimeters.
To reverse the question, ask at what time the depth is 10 centimeters. Press the first button. A copy of the curve turns over the faint diagonal line, which swaps the axes, so the copy shows time against depth. The orange point turns with it, from \((2, 10)\) to \((10, 2)\).
The turned curve is the graph of the inverse function. It takes the depth 10 as its input and returns the time 2.
Swap the input and the output
The curve is now any function \(f\) that rises over its whole domain, and the orange point is \((a, b)\), where \(b = f(a)\).
The mirror point is \((b, a)\). It lies on the graph of the inverse function, which takes \(b\) as its input and returns \(a\).
Drag the point. The values of \(a\) and \(b\) change, but the mirror point is always \((b, a)\), so the inverse takes \(b\) back to \(a\).
Notation
The function \(f\) takes the input \(a\) to the output \(b\).
\[f(a) = \cy{b}\]The inverse function takes \(b\) back to \(a\), so it undoes \(f\).
\[f^{-1}(\cy{b}) = a\]The two statements have the same meaning, read in opposite directions.
\[f^{-1}(b) = a \iff f(a) = b\]The small \(-1\) is a name, not a power, so \(f^{-1}(b)\) is not the reciprocal of \(f(b)\).
\[f^{-1}(b) \neq \frac{1}{f(b)}\]To find \(f^{-1}(b)\), search for the input whose output is \(b\). You can read it from a table or a graph, or solve \(f(a) = b\).
This works if \(f\) is one-to-one, which means that no two inputs have the same output. Then \(f^{-1}\) is a function.
A curve that only rises, or only falls, is one-to-one.
When it fails
The line at height 0 meets the curve \(f(x) = x^3 - 4x\) at three points, with the inputs \(-2\), 0 and 2. All three inputs have the output 0.
Then \(f^{-1}(0)\) would need three answers, but a function gives only one. The line at height 5 meets the curve once, but the height 0 alone means that this \(f\) has no inverse on its whole domain.
Example 1: Read a table backward
A function \(f\) has \(f(1) = 4\), \(f(3) = 9\), \(f(5) = 13\) and \(f(8) = 20\). Find \(f^{-1}(13)\).
The inverse at 13 is the input that \(f\) sends to 13. Search the outputs for 13 with a line at height 13.
The line meets the point whose output is 13, above the input 5. This means \(f(5) = 13\).
\[f(5) = \cy{13}\]Read the statement backward. Because \(f\) takes 5 to 13, the inverse takes 13 back to 5, and the mirror point is \((13, 5)\).
\[f^{-1}(13) = \cy{5}\]
Example 2: Solve for the input
Find \(f^{-1}(12)\) for \(f(x) = x^3 + 2x\).
A tempting wrong move: Read the \(-1\) as a power, and take the reciprocal of \(f(12)\).
\[f^{-1}(12) = \cr{\frac{1}{f(12)} = \frac{1}{1752}}\]The red line marks the input 12, where \(f(12) = 1752\) is far above the picture. The curve passes through \((2, 12)\), so \(f(2) = 12\), and the inverse at 12 must be the input 2, not a tiny fraction.
The \(-1\) names the inverse. The inverse at 12 is the input \(a\) that \(f\) sends to 12, so solve this equation.
\[a^3 + 2a = 12\]Try small whole numbers. The input 1 gives 3, which is too small, and the input 2 gives 12.
\[f(1) = 3, \qquad f(2) = \cy{12}\]The slope \(3x^2 + 2\) is positive, so \(f\) only rises and no other input has the output 12. The mirror point is \((12, 2)\).
\[f^{-1}(12) = \cy{2}\]
Practice
Find \(f^{-1}(18)\) for \(f(x) = x^3 + 5x\).
Move the line up the graph and find the input where the curve reaches height 18. Try small whole numbers.
Solution
The inverse at 18 is the input \(a\) with \(f(a) = 18\). Try \(a = 1\). The output 6 is too small.
\[f(1) = 1 + 5 = 6\]Try \(a = 2\). The orange point reaches height 18 at the input 2.
\[f(2) = 8 + 10 = \cy{18}\]The slope \(3x^2 + 5\) is positive, so \(f\) only rises and no other input has the output 18. The mirror point is \((18, 2)\).
\[f^{-1}(18) = \cy{2}\]
Questions
The figure shows \(f(x) = x^3 + x\), a curve that only rises. The orange point marks the input and the output that each question asks about.
What is \(f^{-1}(2)\)?
- 1
- 2
- \(\tfrac12\)
- 10
Answer
1
Which statement is true?
- \(f^{-1}(10) = 2\)
- \(f^{-1}(2) = 10\)
- \(f^{-1}(10) = \tfrac{1}{10}\)
Answer
\(f^{-1}(10) = 2\)
Let \(f(x) = 2^x + x\), a function that only rises. What is \(f^{-1}(11)\)?
- 3
- 11
- \(\tfrac{1}{11}\)
Answer
3
Summary
You find \(f^{-1}(b)\) by searching a table, a graph or an equation for the input \(a\) whose output is \(b\).
This works because the inverse undoes \(f\), so the point \((a, b)\) on \(f\) becomes the point \((b, a)\) on the inverse.
Use it when a question gives an output of f and asks for the input.
This is one of four skills in topic 3.3 of Which & Why: AP Calculus.