Operations on functions
Add, subtract, multiply or divide two functions output by output; compose them by feeding one function's output into the other.
A worked example
f(x) = x + 1 and g(x) = x² − 2. (f + g)(2) = 3 + 2 = 5. (f ∘ g)(2) = f(g(2)) = f(2) = 3, but (g ∘ f)(2) = g(3) = 7: the order matters.
The mistake people actually make
The domain of f/g leaves out every x where g(x) = 0, even after cancelling. The domain of f ∘ g needs g(x) to be in the domain of f.