Today's soundtrack is *Xzibit: Restless*, bringing some West Coast rap to your ear holes.

This is it. The final lesson of the final unit of the final module of the final high school level math course. In it, I'll be learning about composite functions.

Composite functions exist when one function is *inside* of another function. They can be expressed **"****f**** of ****g**** of ****x****," ****f****(****g****(****x****)), or (****f****∘****g****)(****x****)** - not to be confused with (*f⋅g*)(*x*)!

When working with composite functions, we must follow these steps:

Evaluate

*g*'s outputUse the output of

*g*to evaluate*f*by substituting*g*'s output for*f*'s variableVerify that the range of

*g*includes the domain of*f*

To clarify the third point, if *f* has a restriction that prevent any of *g*'s outputs from being used as an input, it is not a composite function. This is logical enough; we use *g*'s outputs (its *y*-values) as *f*'s inputs (its *x*-values).

We can evaluate compositions by substituting the given value in for *x* on the inner function, simplifying, then substituting that simplified expression into the outer function's variable.

If we are asked to find a function given a composition, we must work in reverse, asking ourselves what values could be substituted for *x* that would give us the final given outcome.

Some functions are inverses of each other. This means that *f*(*g*(*x*)) = *x*, and *g*(*f*(*x*)) = *x*, meaning that after we evaluate both functions and simplify, we are left with only a single variable on either side for both functions.

And with that, I'm finished this course. All that's left is to study for my final exam.