Use inverse information from data
Find the row by its output
The table gives a function \(f\) and its slope \(f'\) at four inputs, one input in each row.
To find the slope of the inverse at 5, search the column of \(f(x)\) for 5. It is in the row of the input 2, so 2 is the matching input, the input that \(f\) sends to 5.
In the same row, the slope of \(f\) is 0.5, so the slope of the inverse at 5 is the reciprocal, 2.
The last button marks in red the slope at the input 5. The inverse at 5 does not use it, because it belongs to a different point of the curve.
Find the row, then take the reciprocal
A table gives \(f\) and its slope at the same inputs, and you want the slope of the inverse at an output \(b\).
Move the slider to put the orange outline on the row whose output is \(b\). The input in that row is \(a\).
Whichever row it is, you take the reciprocal of the slope in that row. The slope is read at the input \(a\), not at the output \(b\).
Notation
Find the row whose output is \(b\). The input in that row is \(a\): for \(b = 10\), it is \(a = 8\).
\[f(a) = \cy{b}\]Read the slope of \(f\) in the same row.
\[\cy{f'(a)}\]Take the reciprocal. This is the slope of the inverse at \(b\).
\[(f^{-1})'(b) = \frac{1}{\cy{f'(a)}}\]The slope of \(f\) is in output units per input unit, so the slope of the inverse is in input units per output unit.
The table must contain a row with the output \(b\). The slope \(f'(a)\) must not be 0, because you cannot divide by 0.
When it fails
No row has the output 6, so the table does not give the matching input. The nearest outputs, 5 and 7, have the slopes 0.5 and 0.8, and neither is the slope at 6.
Do not use a nearby row instead. Find more data, or say that the table does not give enough information.
Example 1: Read a graph
The graph shows an increasing function \(f\). The tangent at the point \((6, 4)\) passes through \((3, 0)\). Find \((f^{-1})'(4)\).
The dashed line at height 4 meets the curve at \((6, 4)\), so the matching input is 6.
\[f(6) = \cy{4}\]The tangent passes through \((3, 0)\) and \((6, 4)\), so it rises 4, shown in green, over a run of 3.
\[f'(6) = \frac{4 - 0}{6 - 3} = \cy{\frac43}\]Take the reciprocal. The graph of \(f^{-1}\) is the mirror image of the curve across the diagonal.
\[(f^{-1})'(4) = \frac{1}{f'(6)} = \cy{\frac34}\]The tangent at \((6, 4)\) is steeper than the diagonal, so the mirror tangent at \((4, 6)\) is less steep, which matches a slope below 1.
Example 2: Read a table
A table gives \(f(x) = 6, 9, 12, 14\) and \(f'(x) = 1.8, 1.2, 0.75, 0.5\) at the inputs \(x = 4, 6, 9, 12\). Find \((f^{-1})'(9)\).
A tempting wrong move: The inverse is taken at 9, so use the row of the input 9.
\[\begin{aligned} (f^{-1})'(9) &= \cr{\frac{1}{f'(9)}} \\ &= \cr{\frac{1}{0.75} = \frac43} \end{aligned}\]The row of the input 9 has the output 12, so the red cell is the slope at the point \((9, 12)\). The inverse at 9 needs the point whose output is 9.
Search the column of \(f(x)\) for 9. It is in the row of the input 6, so the matching input is 6.
\[f(6) = \cy{9}\]Read the slope in that row. Then take the reciprocal.
\[\begin{aligned} f'(6) &= 1.2 \\ (f^{-1})'(9) &= \frac{1}{1.2} = \cy{\frac56} \end{aligned}\]
Practice
A town has \(P(t)\) thousand people, \(t\) years after the year 2000. Given \(P(10) = 52\) and \(P'(10) = 1.6\), find \((P^{-1})'(52)\) and say what it means.
The sketch passes through the only point that the data give. The inverse takes a population and returns a year, so its slope is in years per thousand people.
Solution
The data give \(P(10) = 52\), so the matching input is 10.
\[a = \cy{10}\]Take the reciprocal of the slope of \(P\) at 10.
\[\begin{aligned} (P^{-1})'(52) &= \frac{1}{P'(10)} \\ &= \frac{1}{1.6} = \cy{0.625} \end{aligned}\]The units are years per thousand people, so near 52 thousand, each additional thousand people takes about 0.625 years.
Questions
The graph shows \(f(x) = 3\sqrt{x}\). Its slope is \(\frac32\) at \((1, 3)\) and \(\frac34\) at \((4, 6)\).
What is \((f^{-1})'(6)\)?
- \(\tfrac43\)
- \(\tfrac34\)
- \(\tfrac{2\sqrt6}{3}\)
- \(\tfrac23\)
Answer
\(\tfrac43\)
At which output \(b\) is the slope of the inverse \(\frac23\)?
- \(b = 3\)
- \(b = 1\)
- \(b = 6\)
- \(b = \tfrac23\)
Answer
\(b = 3\)
A table gives \(f(1) = 3\), \(f(3) = 6.2\), \(f(6) = 10\) and \(f(10) = 14\), and \(f'(1) = 1.8\), \(f'(3) = 1.6\), \(f'(6) = 1.25\) and \(f'(10) = 0.5\). What is \((f^{-1})'(10)\)?
- 0.8
- 1.25
- 2
- 0.5
Answer
0.8
Summary
You find the row or the point where the output is \(b\), read the slope of \(f\) at that input, and take the reciprocal.
This works because the inverse is the mirror image of \(f\), so a slope at \((a, b)\) becomes the reciprocal slope at \((b, a)\).
It needs no formula, so it works for a table, a graph or a few given values of \(f\) and its slope.
This is one of four skills in topic 3.3 of Which & Why: AP Calculus.