Composite and inverse functions: FAQ

Composing functions allows us to combine two or more functions into one. We can use this technique to build more complicated functions from simpler ones. It also allows us to model more complex situations in math, physics, engineering, and other disciplines.

An invertible function is one that can be "undone" or reversed. In other words, we can find another function that will take us back to where we started. For example, a function that adds $2$‍ to a number is invertible: the inverse function would subtract $2$‍ from a number.

In general, we don't always need to restrict the domain of a function to make it invertible. However, there are certain cases where it is necessary.

To be invertible, a function must have the property that each input maps to a unique output. Otherwise, the inverse function won't be able to "undo" the original function unambiguously.

For example, in the function $f(x)=x^2$‍, both $-2$‍ and $2$‍ map to $4$‍. Should the inverse function map $4$‍ back to $2$‍ or $-2$‍? It's ambiguous, but a function can only have one output for a given input.

However, if we restrict the domain of $f$‍ to $x\ge 0$‍ (non-negative numbers), then the function becomes invertible.

The easiest way to graph an inverse function is by reflecting the original function across the line $y=x$‍. This is because the inputs and outputs switch places when we invert a function.

These functions are used in a lot of places! For example, in physics, we might use a composite function to model the motion of a falling object. In economics, we might use an inverse function to model the relationship between supply and demand.