# 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 $$f.$$