How to Find the Inverse of the Function f(x) = (x-4)/(4x+5)

Finding the Inverse of the Function f(x) = (x - 4) / (4x + 5)

In this tutorial, we'll walk through the process of finding the inverse of the function f(x) = (x - 4) / (4x + 5). Follow these steps:

Step 1: Replace f(x) with y

Begin by replacing the function notation f(x) with a simple 'y':

y = (x - 4) / (4x + 5)

Step 2: Swap x and y

Next, swap the positions of 'x' and 'y' in the equation:

x = (y - 4) / (4y + 5)

Step 3: Solve for y

Now, we need to isolate 'y' on one side of the equation. Here's how:

Multiply both sides of the equation by (4y + 5):
```
x(4y + 5) = y - 4 
```
Distribute 'x' on the left side:
```
4xy + 5x = y - 4
```
Move all terms with 'y' to one side and all other terms to the other side:
```
4xy - y = -5x - 4
```
Factor out 'y' on the left side:
```
y(4x - 1) = -5x - 4
```
Divide both sides by (4x - 1) to isolate 'y':
```
y = (-5x - 4) / (4x - 1)
```

Therefore, the inverse function of f(x) is:

f^(-1)(x) = (-5x - 4) / (4x - 1)

This tutorial provides a clear and concise explanation of how to find the inverse of the function f(x) = (x - 4) / (4x + 5). By following these steps, you can successfully determine the inverse function.