Problem 71 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \( f(x) = 2x + 3 \).

Step 2: 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 \( x = 2y + 3 \).

Step 3: 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 \( y = (x - 3) / 2 \).

Step 4: 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.