Composite Functions (Function Composition)
1. What Is Function Composition?
Suppose you have two functions, $f$ and $g$. Composition means you apply one function to the result of the other. The standard notation is:
$$ (f \circ g)(x) = f\!\bigl(g(x)\bigr) $$We read this as "f composed with g at x" or simply "f of g of x." The symbol $\circ$ is called the composition operator.
Which function acts first?
We follow the usual convention with parentheses by starting with the innermost parentheses first:
- The inner function $g$ takes the input $x$ first and produces $g(x)$.
- The outer function $f$ then takes $g(x)$ as its input and produces $f(g(x))$.
Think of it as a two-stage machine:
$$ x \;\xrightarrow{\;g\;}\; g(x) \;\xrightarrow{\;f\;}\; f\!\bigl(g(x)\bigr) $$Real-world analogy: Suppose the temperature on day $d$ of the year is $T(d)$, and the heating cost at temperature $T$ is $C(T)$. Then the cost on day $d$ is $C\!\bigl(T(d)\bigr)$ — a composition where $T$ acts first (inner) and $C$ acts second (outer).
2. Evaluating Compositions & Building Formulas
Evaluating at a Specific Point
To evaluate $f(g(a))$ for a specific number $a$, work inside-out:
- Compute $g(a)$ — get a number.
- Substitute that number into $f$.
Let $f(x) = x^2 - 2x$ and $g(x) = x - 5$. Find $f(g(-1))$.
Step 1: $g(-1) = -1 - 5 = -6$