Inverse of a Rational Function

To find the inverse of a rational function y = f(x):

• Replace f(x) with y.
• Interchange x and y.
• Solve the resultant equation for y.
• The result would give the inverse f^-1(x).

Example: Find the inverse of the rational function f(x) = (2x – 1) / (x + 3).

Solution:

The given function can be written as:

y = (2x – 1) / (x + 3)

Interchanging x and y:

x = (2y – 1) / (y + 3)

Now, we will solve for y.

x(y + 3) = 2y – 1

xy + 3x = 2y – 1

3x + 1 = 2y – xy

3x + 1 = y (2 – x)

y = (3x + 1) / (2 – x) = f^-1(x)