Function Composition
A free Algebra II lesson from the “Functions and Notation” unit, with a worked example and practice problems including step-by-step solutions.
Function composition means using the output of one function as the input of another. In f(g(x)), the inside function g is evaluated first, and its result is placed into f. Composition matters because many real situations happen in stages, such as applying a discount and then tax or converting units twice. When practicing, work from the inside outward and write each substitution clearly. A common mistake is multiplying f and g or evaluating the outside function first. Always identify the inside function before simplifying.
What you'll learn
- Evaluate composite functions
- Track inside and outside rules
- Use composition in context
Worked example
Problem. If f(x) = 2x + 1 and g(x) = x - 4, find f(g(10)).
- Evaluate inside first: g(10) = 6.
- Now find f(6).
- 2(6) + 1 = 13.
Answer: 13
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. If f(x) = x + 3 and g(x) = 2x, find f(g(4)).
Show solution
- g(4) = 8.
- f(8) = 11.
Answer: 11
2. If f(x) = 5x and g(x) = x - 1, find g(f(2)).
Show solution
- f(2) = 10.
- g(10) = 9.
Answer: 9
3. If f(x) = x^2 and g(x) = x + 2, find f(g(3)).
Show solution
- g(3) = 5.
- f(5) = 25.
Answer: 25
4. If f(x) = 3x - 7 and g(x) = x^2, find f(g(4)).
Show solution
- g(4) = 16.
- 3(16) - 7 = 41.
Answer: 41
5. For f(g(x)), which function happens first?
Choices: g · f · Both at once · Neither
Show solution
- The inside function is evaluated first.
Answer: g
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