Combining functions

Definition

Combining functions involves creating a new function by applying one function to the results of another. This is often done through operations such as addition, subtraction, multiplication, division, and composition.

5 Must Know Facts For Your Next Test

1. The composition of two functions $f$ and $g$ is denoted as $(f \circ g)(x)$ or $f(g(x))$.
2. For composition to be defined, the range of the inner function must be within the domain of the outer function.
3. Composition is not necessarily commutative; that is, $f(g(x))$ may not equal $g(f(x))$.
4. To find $(f \circ g)(x)$, first apply $g(x)$ and then apply $f$ to that result.
5. The domain of $(f \circ g)(x)$ consists of all $x$ in the domain of $g$ for which $g(x)$ lies in the domain of $f$.

Review Questions

• What are the necessary conditions for composing two functions?
• Is it always true that $(f \circ g)(x) = (g \circ f)(x)? Why or why not?
• How do you determine the domain of a composed function?

Related terms

Function: A relation between a set of inputs and a set of permissible outputs where each input is related to exactly one output.

Domain: The set of all possible inputs for which a function is defined.

Range: $The set of all possible outputs produced by a function.$