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Chapter 2: Problem 93

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\text { If } f(x)=3 x, \text { then } f^{-1}(x)=\frac{1}{3 x}$$

Short Answer

Expert verified
The given statement is false. The true inverse of the function \(f(x) = 3x\) is \(f^{-1}(x) = \frac{x}{3}\), not \(f^{-1}(x) = \frac{1}{3x}\).

Step by step solution

01

Identifying the Function and Its Proposed Inverse

Begin by identifying the function and its proposed inverse. The function given here is \(f(x) = 3x\) and the proposed inverse is \(f^{-1}(x) = \frac{1}{3x}\).
02

Find the Real Inverse of the Function

To find the true inverse of a function \(f(x)\), we need to swap \(x\) and \(y\), and then solve for \(y\). Therefore, if \(y = 3x\), swapping \(x\) and \(y\) gives \(x = 3y\). Solving for \(y\) gives \(y = \frac{x}{3}\). This shows that the actual inverse function is \(f^{-1}(x) = \frac{x}{3}\), and not \(f^{-1}(x) = \frac{1}{3x}\).
03

Conclusion

The statement is false. The true inverse of the function \(f(x) = 3x\) is \(f^{-1}(x) = \frac{x}{3}\), and not \(f^{-1}(x) = \frac{1}{3x}\).

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