Composition of Functions

Definition

The composition of functions, denoted as $f \circ g$ (read as “f composed with g” or “f of g”), is defined as applying one function to the result of another. Mathematically:

$$(f \circ g)(x) = f(g(x))$$

In simpler terms, you first apply the function $g$ to $x$, and then you apply the function $f$ to the result, $g(x)$.

Condition for Composition

For the composition $f(g(x))$ to be defined, the range of $g$ must be a subset of the domain of $f$.

$$r_g \subseteq d_f$$

This ensures that the output of $g(x)$ is a valid input for $f(x)$. If this condition is not met, the composite function is not defined for all $x$ in the domain of $g$.

Domain of Composite Function

The domain of the composite function $f \circ g$ is the same as the domain of the inner function $g$, provided the range condition is met.

$$dom(f \circ g) = dom(g)$$

Evaluating Composite Functions

1. Find the rule for the composite function: Substitute $g(x)$ into $f(x)$ wherever you see $x$ in the expression for $f(x)$.
2. Determine the domain: The domain of $f(g(x))$ is the set of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.
3. Determine the range: After finding the rule and domain, determine the range of the composite function.

Example

Let $f(x) = x^2$ and $g(x) = x + 1$.

1. Find $(f \circ g)(x)$:
   $$(f \circ g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1$$

2. Find $(g \circ f)(x)$:
   $$(g \circ f)(x) = g(f(x)) = g(x^2) = x^2 + 1$$

Notice that in general, $f \circ g \neq g \circ f$.

Domain and Range Example

Let $f: \mathbb{R} \rightarrow \mathbb{R}$, $f(x) = 2x - 1$ and $g: \mathbb{R} \rightarrow \mathbb{R}$, $g(x) = 3x^2$.

• Find $f \circ g$:
  $$f(g(x)) = f(3x^2) = 2(3x^2) - 1 = 6x^2 - 1$$
  $dom(f \circ g) = \mathbb{R}$
  $ran(f \circ g) = [-1, \infty)$

• Find $g \circ f$:
  $$g(f(x)) = g(2x - 1) = 3(2x - 1)^2 = 3(4x^2 - 4x + 1) = 12x^2 - 12x + 3$$
  $dom(g \circ f) = \mathbb{R}$
  $ran(g \circ f) = [0, \infty)$

Key Points

• The order of composition matters: $f(g(x))$ is generally different from $g(f(x))$.
• The domain of the composite function depends on both the domain of the inner function and the condition $r_g \subseteq d_f$.
• To find the rule for a composite function, substitute the inner function into the outer function.

Table of Composite Functions