Question

Find the inverse function of \(f\). Use a graphing utility to graph \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$ f(x)=\frac{x}{\sqrt{x^{2}+7}} $$

Short answer

The inverse function of \(f(x) = \frac{x}{\sqrt{x^{2}+7}}\) is \(f^{-1}(x) = \sqrt{\frac{7x^{2}}{1-x^{2}}}\). The graph of \(f^{-1}(x)\) will be a reflection of the graph of \(f(x)\) across the line \(y=x\), demonstrating symmetry about the line \(y=x\).

Step-by-step solution

Finding the Inverse Function

The inverse function of \(f\) is found by replacing \(f(x)\) with \(y\), swapping \(x\) and \(y\), and then solving the resulting equation for \(y\). So, \[y = \frac{x}{\sqrt{x^{2}+7}}\] Swap \(x\) and \(y\): \[x= \frac{y}{\sqrt{y^{2}+7}}\] Cross multiply: \(x\sqrt{y^{2}+7} = y\) Square both sides: \(x^{2}(y^{2}+7) = y^{2}\] Rearrange terms: \(x^{2}y^{2} + 7x^{2} = y^{2}\] Combine like terms: \((x^2 - 1)y^{2}= -7x^{2}\] Divide by \((x^2 - 1)\) and multiply by -1: \[y^{2} = \frac{7x^{2}}{1-x^{2}}\] Since \(y \(>=0\) because it's a square root of \(x\), the inverse function is \(f^{-1}(x) = \sqrt{\frac{7x^{2}}{1-x^{2}}}\) or \(f^{-1}(x) = -\sqrt{\frac{7x^{2}}{1-x^{2}}}\). We discard the negative solution in this context.

Graphing the Function \(f\) and the Inverse Function \(f^{-1}\)

Use a graphing utility to plot the original function \(f(x) = \frac{x}{\sqrt{x^{2}+7}}\) and the inverse function \(f^{-1}(x) = \sqrt{\frac{7x^{2}}{1-x^{2}}}\) in the same viewing window. One can use any graphing software or tools such as Desmos, GeoGebra, or even the graphing functions on a standard scientific calculator.

Observing the Relationship between the Graphs

After graphing the function \(f\) and its inverse \(f^{-1}\), observe how the two graphs relate to each other. The graph of the inverse function \(f^{-1}\) is a reflection of the original function \(f\) across the line \(y=x\). Both graphs are symmetrical about the line \(y=x\). This is a standard property of functions and their inverses.