Composite functions?

Question

What is a composite function, (inside/outside), and rules for range and domain?

Answer 1

If a function is like a machine, then a composite function is like a composite machine: the output of the first one becomes the input of the second.

f(g(x)) means "take x, run it through the g function, take the result, and run it through the f function". There is a complication if the output (range) of g extends over values which are not allowed inputs (domain) for f. In this case you have to restrict the input values of x so that the output of the first stage fits into the second.

For example, consider f(whatever)= 1/(whatever-1) and g(whatever)=whatever +3.

Then f(g(x))=f(x+3)=1/(x+3-1)=1/(x+2). Now this has a big problem when x=-2, so we have to disallow x=-2 as an input to the machine because even though g(2) is defined, f(g(2)) is not.