A composite function is formed when the output of one function becomes the input of another function. It is written as f(g(x)) and read as f of g of x. Composite functions matter because many real situations happen in stages, such as converting units and then calculating cost, or finding a radius and then calculating area.
Thinking of composition as a function machine pipeline helps make the order clear.
Key Facts
- A composite function puts one function inside another: (f o g)(x) = f(g(x)).
- The inner function is evaluated first: in f(g(x)), calculate g(x) before applying f.
- Order usually matters: f(g(x)) is not always equal to g(f(x)).
- To evaluate numerically, substitute the input into g first, then substitute that result into f.
- The domain of f(g(x)) includes values of x that are in the domain of g and make g(x) fall in the domain of f.
- Example: if f(x) = 2x + 1 and g(x) = x^2, then f(g(x)) = 2x^2 + 1.
Vocabulary
- Composite function
- A function made by using the output of one function as the input of another function.
- Inner function
- The function evaluated first in a composite expression such as g in f(g(x)).
- Outer function
- The function applied after the inner function, such as f in f(g(x)).
- Domain
- The set of input values for which a function or composite function is defined.
- Function notation
- A way to name and evaluate functions using symbols such as f(x), g(x), and f(g(x)).
Common Mistakes to Avoid
- Evaluating the outer function first is wrong because f(g(x)) means g(x) must be found before f is applied.
- Assuming f(g(x)) = g(f(x)) is wrong because changing the order often changes the final expression or value.
- Forgetting parentheses during substitution is wrong because expressions like f(x) = x^2 and g(x) = x + 3 require f(g(x)) = (x + 3)^2, not x + 3^2.
- Ignoring domain restrictions is wrong because even if g(x) is defined, the value g(x) must also be allowed as an input to f.
Practice Questions
- 1 Let f(x) = 3x - 4 and g(x) = x^2 + 1. Find f(g(2)) and g(f(2)).
- 2 Let f(x) = sqrt(x) and g(x) = x - 5. Find the domain of f(g(x)).
- 3 Explain why the order of composition matters by comparing f(g(x)) and g(f(x)) for f(x) = x + 2 and g(x) = 4x.