Problem 48 - College Algebra

Question

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

Step-by-Step Solution

Answer

The inverse of the function \( f(x) = 2x + 3 \) is \( f^{-1}(x) = (x - 3) / 2 \). The general method to find the inverse of a one-to-one function includes identifying the function, swapping x and y variables, solving for the new y, and then finally writing the inverse function.

Step 1. Identify the Function

First of all, identify the function for which you need to find the inverse. Let's call this function \( f(x) \). Suppose our one-to-one function is \( f(x) = 2x + 3 \). From now on, we’ll be working with this specific function as an example, although the method explained remains the same for any one-to-one function.

Step 2. Swap x and y variables

An important step in finding the inverse of the function is swapping the variables x and y. This is based on the idea that if \( f(x) = y \), then the inverse function at y, often noted as \( f^{-1}(y) \), will be x. So we swap x and y in our example, changing the equation to \( x = 2y + 3 \).

Step 3. Solve for New y

Now, solve the equation from step 2 for the new y, which will represent our \( f^{-1}(x) \). In our example, we subtract 3 from both sides to isolate the term with y and then divide by 2 to isolate y itself. Our resulting equation is \( y = (x - 3) / 2 \).

Step 4. Write the Inverse Function

Write the inverse function, \( f^{-1}(x) \), replacing \( y \) with \( f^{-1}(x) \). Therefore, the inverse of the function \( f(x) = 2x + 3 \) is \( f^{-1}(x) = (x - 3) / 2 \).