Problem 71 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

To find the inverse of a one-to-one function, you need to confirm the function is one-to-one, swap the variables, solve the equation for \( y \), and then confirm your solution by substituting back into the original equation.

Step 1: Define a One-to-One Function

A one-to-one function, often written as 1-1, is a function where every element of the range corresponds to exactly one element of the domain. So, it is crucial to confirm whether the function you want its inverse is one-to-one.

Step 2: Swap the Variables

Once you have confirmed the function is one-to-one, the next step is to swap the variables. If your function is written in the form \(f(x) = y\), rewrite it in the form \(f(y) = x\). The primary reason for this is that, by definition, the inverse of a function \(f\), denoted as \(f^{-1}\), satisfies \(f(y) = x\) if and only if \(f^{-1}(x) = y\).

Step 3: Solve for y

Now that you have swapped the variables to make \(x\) the subject of the equation, the next goal is to make \(y\) the subject. To do this, perform whichever operations are necessary based on the specific equation you are working with.

Step 4: Confirm the Result

It is crucial to confirm whether your solution is correct. You can do this by substituting your solution into your original equation and checking if the resulting expression for \(y\) is identical to the original form of the equation.