How to Find the Inverse of a Function with a Fraction?

To find the inverse of a function with a fraction, interchange the function’s input and output variables and solve for the original input variable in terms of the original output variable.

Finding the inverse of a function involves swapping the input and output variables and solving for the original input variable in terms of the original output variable.

Steps to find the inverse of a function

Replace the function notation with y.

Swap the x and y variables.

Solve for y to express the inverse function.

If the expression involves fractions, handle them accordingly by simplifying or rationalizing the denominator if necessary.

Verify the obtained inverse function by checking if composing it with the original function results in x.

Here’s a detailed explanation, assuming you have a function y = f(x) with a fraction:

• Original Function: Start with the given function y=f(x) that contains a fraction, for example, y = (2x+3)/5​.

• Swap Variables: Replace y with x and x with y to obtain the equation x = (2y+3)/5​​.

• Solve for the Original Input Variable:

Solve the equation obtained in step 2 for y in terms of x. In the example:

x = (2y+3)/5

Multiply both sides by 5 to clear the fraction:

5x = 2y+3

Subtract 3 from both sides:

5x−3 = 2y

Divide by 2:
y = (2x+3)/5​

• Inverse Function: The result, y = (2x+3)/5​ represents the inverse function of the original function y = (2x+3)/5​.

• Check the Inverse: Verify the inverse by composing the functions. If f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, then the functions are inverses of each other.

This process ensures that you have found the inverse function, taking into account the fraction in the original function. The key is to interchange the variables and solve for the original input variable in terms of the original output variable.