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Chapter 0: Problem 30

Find the inverse of \(f .\) Then use a graphing utility to plot the graphs of \(f\) and \(f^{-1}\) using the same viewing window. $$ f(x)=1-\frac{1}{x} $$

Short Answer

Expert verified
The inverse function of \(f(x) = 1 - \frac{1}{x}\) is \(f^{-1}(x) = \frac{1}{x - 1}\). Use a graphing utility to plot both functions in the same viewing window.

Step by step solution

01

Solution for the inverse function

To find the inverse function, let \(f^{-1}(x)\) denote the inverse of function \(f(x)\). First, replace \(f(x)\) with \(y\): \[ y = 1-\frac{1}{x} \] Next, swap the variables \(x\) and \(y\): \[ x = 1-\frac{1}{y} \] Now, we solve for the variable \(y\).
02

Solving for y

First, isolate the fraction on one side of the equation: \[ x - 1 = -\frac{1}{y} \] Now, we can take the reciprocal of both sides to free the \(y\) variable from the fraction: \[ \frac{1}{x - 1} = y \]
03

Write the inverse function

We now have the expression for \(y\) in terms of \(x\), which represents the inverse function of \(f(x)\). Write the found inverse function in proper notation: \[ f^{-1}(x) = \frac{1}{x - 1} \]
04

Graph the functions using a graphing utility

Now that we have obtained the inverse function, use a graphing utility to plot the original function \(f(x) = 1 - \frac{1}{x}\) and its inverse function \(f^{-1}(x) = \frac{1}{x - 1}\) in the same viewing window.

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Most popular questions from this chapter

Find \(f^{-1}(a)\) for the function \(f\) and the real number \(a\). $$ f(x)=\frac{3}{\pi} x+\sin x ; \quad-\frac{\pi}{2}

Show that \(f\) and \(g\) are inverses of each other by verifying that \(f[g(x)]=x\) and \(g[f(x)]=x\). $$ f(x)=2 x+3 ; \quad g(x)=\frac{x-3}{2} $$

Find the zero(s) of the function f to five decimal places. $$ f(x)=\sin 2 x-x^{2}+1 $$

Let \(f(x)=2 x^{3}-5 x^{2}+x-2\) and \(g(x)=2 x^{3}\). a. Plot the graph of \(f\) and \(g\) using the same viewing window: \([-5,5] \times[-5,5]\). b. Plot the graph of \(f\) and \(g\) using the same viewing window: \([-50,50] \times[-100,000,100,000] .\) c. Explain why the graphs of \(f\) and \(g\) that you obtained in part (b) seem to coalesce as \(x\) increases or decreases without bound. Hint: Write \(f(x)=2 x^{3}\left(1-\frac{5}{2 x}+\frac{1}{2 x^{2}}-\frac{1}{x^{3}}\right)\) and study its behavior for large values of \(x\).

a. Show that \(f(x)=-x^{2}+x+1\) on \(\left[\frac{1}{2}, \infty\right)\) and \(g(x)=\frac{1}{2}+\sqrt{\frac{5}{4}-x}\) on \(\left(-\infty, \frac{5}{4}\right)\) are inverses of each other. b. Solve the equation \(-x^{2}+x+1=\frac{1}{2}+\sqrt{\frac{5}{4}-x}\). Hint: Use the result of part (a).
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