An inverse function, written as f⁻¹(x), is a function that 'undoes' the original function. It's a reverse switch.

If your original function machine turns a '3' into a '10', the inverse function machine turns a '10' back into a '3'.

There is a simple, 4-step recipe for finding the inverse of any function.

Step 1: Rewrite your function, but replace the 'f(x)' with 'y'. So, f(x) = 2x + 3 becomes y = 2x + 3.

Step 2: The Magic Step. Swap every 'x' with a 'y' and every 'y' with an 'x'. So, y = 2x + 3 becomes x = 2y + 3.

Step 3: Solve for the new 'y'. Use your algebra skills to get the new 'y' all by itself on one side.

Step 4: Replace the new 'y' with the proper inverse notation, f⁻¹(x).

Geometrically, the graph of a function and its inverse are perfect mirror images across the line y=x.

Not all functions have a proper inverse. A function must be 'one-to-one' (pass the horizontal line test) for its inverse to also be a function.

This process of 'swapping and solving' is a powerful algebraic technique for reversing any process.