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Worksheet Derivatives of Inverse Functions - Activity 2.6.5

Activity 33.

The graph of \(f(x)=x^3+x+1\) is shown in Figure 161.
Graph of a smooth increasing curve y = f(x) with a labeled point at (1, 3). The point is marked on the curve, which rises steeply for x greater than 1 and more gradually for x less than 1.
Figure 161. The graph of \(f(x)=x^3+x+1\text{.}\)

(a)

Calculate the value of \(f^{-1}(3)\text{.}\)
Hint.
\(f(1)=?\)
Answer.
\(f^{-1}(3)=1\text{.}\)

(b)

Plot and label the point \((3,f^{-1}(3))\text{.}\)
Answer.
The point \((3,1)\) is the reflection of the point \((1,3)\) across the line \(y=x\text{.}\)

(c)

Sketch the graph of \(f^{-1}(x)\) on the grid given in Figure 161.
Answer.
Note that the graph of \(f^{-1}(x)\) is a reflection about \(y=x\text{.}\)
Graph of a blue function y = f(x) that passes through the point (1, 3), shown with the dashed line y = x and the red dashed graph of its inverse passing through (3, 1). The two curves reflect across the line y = x.
Figure 162. The graph of \(f(x)=x^3+x+1\) in solid blue and the graph of \(f^{-1}(x)\) in dashed red.

(d)

Determine the value of \(f'(1)\text{.}\) Then, sketch the tangent line to \(y=f(x)\) through the point \((1,3)\text{.}\)
Hint.
\(f'(x)=3x^2+1\)
Solution.
\(f'(x)=3x^2+1\) gives \(f'(1)=4\text{.}\) The equation of the tangent line is then \(y-3 = 4(x-1)\) which is plotted in Figure 163.
Graph of a blue function y = f(x) passing through (1, 3), shown with the dashed line y = x and the dashed red graph of its inverse passing through (3, 1). A black tangent line at (1, 3) with equation y = 4x − 1 is also drawn.
Figure 163. The graph of \(f(x)=x^3+x+1\) in solid blue and the graph of \(f^{-1}(x)\) in dashed red.

(e)

Sketch the tangent line to \(y=f^{-1}(x)\) at \(x=3\text{.}\) What is the equation for this tangent line?
Solution.
Since the slope is the reciprocal of the slope to the tangent line to \(f(x)\) at \((1,3)\text{,}\) we compute the tangent line at \(x=3\) as
\begin{equation*} y-1 = \frac{1}{4}(x-3)\text{.} \end{equation*}
Graph of a blue function y = f(x) passing through the point (1, 3), shown with the dashed line y = x and the dashed red graph of the inverse function passing through (3, 1). A black tangent line at (1, 3) with equation y = 4x − 1 and the corresponding reflected line at (3, 1) with equation y = ¼x + ¼ are also drawn.
Figure 164. The graph of \(f(x)=x^3+x+1\) in solid blue and the graph of \(f^{-1}(x)\) in dashed red. Tangent lines at \((1,3)\) and \((3,1)\) to each are shown.

(f)

Fill-in the blanks in the following sentence: The \(\underline{\hspace{20mm}}\) of the tangent line to \(y=f^{-1}(x)\) at \(x=\underline{\hspace{20mm}}\) is the \(x=\underline{\hspace{20mm}}\) of the slope of the tangent line to \(y=f(x)\) at \(x=a\text{.}\) Use items from this list: \(a\text{,}\) \(b\) , \(f(a)\) , \(f(b)\) , concavity , slope , negative , reciprocal
Answer.
slope; \(f(a)\text{;}\) reciprocal
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