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:

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 \(f\).

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.

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