Combining and composing functions
Two functions share an input, or one feeds the other. How do you build a single new function out of the two?
Where you are 5 inside
What this area is about
Functions can be combined into new ones. Two functions with the same input can be added, subtracted, multiplied or divided output by output, giving f + g or \displaystyle \frac{f}{g}. Or one function’s output can be fed into another, giving the composition f(g(x)). Each new function has its own rule and its own domain, and finding that domain is the part that needs care.
How these fit together
Arithmetic of functions. Adding, subtracting and multiplying functions combines outputs at the same input. Dividing functions and the domain of a combination does the same for division, removes the inputs where the bottom output is 0, and finds which inputs any combination accepts.
Composition. Evaluating a composition runs one number through the inner function and then the outer one, from formulas, tables or graphs. Composing functions algebraically substitutes the whole inner rule to get one formula.
The domain of a chain. Domain of a composition says which starting inputs make it all the way through the chain.
A path through it
Sums and products, then quotients. Separately, evaluate a composition at a number, then compose algebraically. The domain of a composition comes last, after both quotients and algebraic composition.
Composition is how inverses are checked, and it returns in calculus as the chain rule.