1.8 | Combinations of Functions: Composite Functions


Let \(f\) and \(g\) be two functions with overlapping domains. Then, for all \(x\) common to both domains, sum, difference, product, and quotient of \(f\) and \(g\) are defined as follows.

  1. Sum:
    $$
    (f+g)(x) = f(x) + g(x)
    $$
  2. Difference:
    $$
    (f-g)(x) = f(x) – g(x)
    $$
  3. Product:
    $$
    (fg)(x) = f(x) \cdot g(x)
    $$
  4. Quotient:
    $$
    \left( \frac{f}{g} \right) (x) = \frac{f(x)}{g(x)}, \quad g(x) \not = 0
    $$
The composition of the function \(f\) with the function \(g\) is
$$
(f \circ g)(x) = f(g(x)).
$$
The domain of \(f \circ g\) is the set of all \(x\) in the domain of \(g\) such that \(g(x)\) is in the domain of \(f.\)

E 1.8 Exercises