Problem 71 - College Algebra 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, start by expressing the function. Write it in single-variable form using y instead of f(x). Then, interchange the variables. Every place you see 'x', replace it with 'y' and vice versa. Following this, solve the equation for 'y'. The expression obtained for 'y' indicates the inverse function of the given function. Finally, check by substitcribing points from the original function to ensure correctness of the inverse function.

Step 1: Express the function

Write the function in y=f(x) form.

Step 2: Interchange The Variables

Swap every x for a y and swap every y for an x. So the function becomes x=f(y).

Step 3: Solve for 'y'

Once the variables have been interchanged, solve the function for 'y' to get y=f^(-1)(x). This gives the inverse of the original function.

Step 4: Check

Substitute some easy-to-calculate points from the original function into the inverse function to ensure the output values match the original input values. This step is important to verify the correctness of the inverse function derived.