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Chapter 1: Problem 68

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=|x-2|, \quad x \leq 2$$

Short Answer

Expert verified
The function \(f(x)=|x-2|\) for \(x \leq 2\) does have an inverse. The inverse function is \(f^{-1}(x) = x + 2\).

Step by step solution

01

Test for Injectivity

With the restricted domain \(x \leq 2\) of the function \(f(x)=|x-2|\), it is a one-to-one function.
02

Find the Inverse

To find the inverse of a function, we switch the roles of \(y\) and \(x\). So, we have \(x = |y - 2|\). Since \(x\) is always positive, \(x = y - 2\). Solving for \(y\), we find \(y = x + 2\), which is the inverse function.
03

Verify Inverse Function

To verify \(f^{-1}(x) = x + 2\) is indeed the inverse, we should check it by substituting it into the original function. The expression should yield \(x\) if substituted correctly.

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Most popular questions from this chapter

The function \(F(y)=149.76 \sqrt{10} y^{5 / 2}\) estimates the force \(F\) (in tons) of water against the face of a dam, where \(y\) is the depth of the water (in feet). (a) Complete the table. What can you conclude from the table? $$\begin{array}{|l|l|l|l|l|l|}\hline y & 5 & 10 & 20 & 30 & 40 \\\\\hline F(y) & & & & & \\\\\hline\end{array}$$ (b) Use the table to approximate the depth at which the force against the dam is \(1,000,000\) tons. (c) Find the depth at which the force against the dam is \(1,000,000\) tons algebraically.

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Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(P\) varies directly as \(x\) and inversely as the square of \(y .\) \(\left(P=\frac{28}{3} \text { when } x=42 \text { and } y=9 .\right)\)

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