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Chapter 2: Problem 71

Describe how to find the inverse of a one-to-one function.

Short Answer

Expert verified
The inverse of the one-to-one function \( f(x) = 2x + 3 \) is \( f^{-1}(x) = (x - 3) / 2 \).

Step by step solution

01

Identify the Function

The first step is identifying the function to be inversed. It is assumed that a one-to-one function is given, such as a line, a logarithm or an exponential function. Let's take, for illustrative purposes, the function \( f(x) = 2x + 3 \).
02

Swap x and y variables

The second step is swapping the variables. This means, wherever there is an x, replace it with y and wherever there is y or f(x), replace it with x. So, \( f(x) = 2x + 3 \) becomes \( f^{-1}(y) = 2y + 3 \), or simply \( x = 2y + 3 \).
03

Solve for the new y

The third step is solving for the new y, meaning we need to isolate y in the equation got at previous step. By simple algebra, the equation becomes \( y = (x - 3) / 2 \).
04

Replace the new y with inverse notation

Finally, replace the new y with inverse notation. Therefore, \( y = (x - 3) / 2 \) becomes \( f^{-1}(x) = (x - 3) / 2 \), which represents the inverse of the original function.

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Most popular questions from this chapter

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