How to Find the Inverse of a Function Using a Number Machine

A function can be visualized as a number machine, which takes a number as an input and processes it into another number as an output.

The inverse of the function can be found by reversing the order of each operation on the number machine, and changing each operation to its opposite.

A Real Example of How to Find the Inverse of a Function Using a Number Machine

Question: What is the inverse function of the function:

Step 1: Draw the function as a number machine, using a series of operations (in the correct order) to process the input into the output.

Step 2: Reverse the order of the operations, so the right-most operation goes on the left, and the left-most operation goes on the right.

Step 3: Replace each operation with its opposite.

Replacing the operations in the number machine with their opposites gives:

The number machine now represents the inverse function.

Step 4: Put an input x into the number machine to see what output, f-1(x), comes out.

The inverse function is given by:

Another Real Example of How to Find the Inverse of a Function Using Algebra

The slider below shows a real example of how to find the inverse of a function using a number machine.
show

Note
WHAT IS AN INVERSE FUNCTION?

An inverse function is itself a function which reverses a function.

If a function f(x) maps an input x to an output f(x)...

... an inverse function takes the output f(x) back to the input x:

An inverse function is denoted f-1(x). It relates an input x to an output f-1(x):