How to Algebraically Find the Inverse of a Function

To find the inverse of a function algebraically, follow these steps:

1. Replace the function notation f(x) with y.

2. Switch the positions of x and y.

3. Solve the new equation for y.

4. Replace y with f^(-1)(x), which is the inverse function of f(x).

5. Check your answer by verifying that the composition of f(x) and f^(-1)(x) gives you x, and vice versa.

Here's an example to illustrate the steps:

Find the inverse function of f(x) = 2x + 3.

Step 1: Replace f(x) with y.

y = 2x + 3

Step 2: Switch the positions of x and y.

x = 2y + 3

Step 3: Solve for y.

x - 3 = 2y

y = (x - 3)/2

Step 4: Replace y with f^(-1)(x).

f^(-1)(x) = (x - 3)/2

Step 5: Check your answer by verifying that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.

f(f^(-1)(x)) = f((x - 3)/2) = 2((x - 3)/2) + 3 = x - 3 + 3 = x

f^(-1)(f(x)) = f^(-1)(2x + 3) = (2x + 3 - 3)/2 = x/2

Since both compositions give x, the inverse function is correct.

Therefore, the inverse function of f(x) = 2x + 3 is f^(-1)(x) = (x - 3)/2.