Function Composition: Step-by-Step Guide
Function composition means using the output of one function as the input of another. (f o g)(x) is the same as f(g(x)). You apply g to x first, and then apply f to that result. The order matters, because f(g(x)) is usually different from g(f(x)).
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Find the inner and outer functions. In f(g(x)), the inner function is g, and the outer function is f. The inner function goes first.\[(f\circ g)(x) = f(g(x))\]
- Put the inner function into the outer one. Wherever you see x in f(x), replace it with the whole expression g(x), in parentheses.\[f(x)\to f\bigl(g(x)\bigr)\]
- Simplify. Multiply out and combine like terms to get a clean expression.\[f(g(x)) = \text{simplified polynomial / expression}\]
- Find the domain. x must be allowed in g, and g(x) must be allowed in f. Leave out any values where that fails.\[\text{dom}(f\circ g) = \{x \in \text{dom}(g)\,:\,g(x)\in \text{dom}(f)\}\]
- Check with a number. Pick a number for x. Find g of it, then f of that result. Compare with your composed expression.\[f(g(c)) \stackrel{?}{=} (f\circ g)(c)\]
Worked example
Given f(x) = 2x + 1 and g(x) = x^2 - 3, find (f o g)(x).
Step 1: The inner function is g(x) = x^2 - 3. The outer function is f(x) = 2x + 1.
Step 2: Replace x in f(x) with g(x): f(g(x)) = 2(x^2 - 3) + 1.
Step 3: Simplify: 2(x^2 - 3) + 1 = 2x^2 - 6 + 1 = 2x^2 - 5.
Step 4: Domain: g works for every real number, and f accepts any number, so the domain is all real numbers.
Step 5: The answer is (f o g)(x) = 2x^2 - 5.
Check: Let x = 2. g(2) = 4 - 3 = 1, and f(1) = 2 + 1 = 3. Using the answer: 2(4) - 5 = 3. They match.
Tips and common mistakes
- Order matters. f(g(x)) and g(f(x)) are usually different, so read the notation carefully.
- Always put g(x) in parentheses when you substitute. It stops mistakes when you multiply out.
- Composition is not multiplication. f(g(x)) does not mean f(x) times g(x).
- If the inner function has a square root or a fraction, make sure its outputs are allowed as inputs for the outer function.
- (f o g)(x) is read 'f of g of x'. The function closest to x goes first.
- Trying one number is a quick way to catch most mistakes.