Composition of Functions
A free Precalculus lesson from the “Transformations and Combinations of Functions” unit, with a worked example and practice problems including step-by-step solutions.
Composition means working from the inside out: find the inner output, then feed it into the outer function. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Evaluate and interpret one function used as the input of another
- Use composition of functions in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. If f(x) = 3x - 1 and g(x) = x + 3, find (f o g)(2).
- Worked Example: First identify exactly what the question is asking: If f(x) = 3x - 1 and g(x) = x + 3, find (f o g)(2).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- First compute g(2) = 5.
- Then compute f(5) = 3(5) - 1.
- The value is 14.
- Check the result by substituting or estimating: the response should match 14 and make sense in the original problem.
Answer: 14
Practice problems
1. If f(x) = 3x - 1 and g(x) = x + 3, find (f o g)(2).
Show solution
- Warm-up: First identify exactly what the question is asking: If f(x) = 3x - 1 and g(x) = x + 3, find (f o g)(2).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- First compute g(2) = 5.
- Then compute f(5) = 3(5) - 1.
- The value is 14.
- Check the result by substituting or estimating: the response should match 14 and make sense in the original problem.
Answer: 14
2. If f(x) = 4x - 1 and g(x) = x + 3, find (f o g)(3).
Show solution
- Warm-up: First identify exactly what the question is asking: If f(x) = 4x - 1 and g(x) = x + 3, find (f o g)(3).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- First compute g(3) = 6.
- Then compute f(6) = 4(6) - 1.
- The value is 23.
- Check the result by substituting or estimating: the response should match 23 and make sense in the original problem.
Answer: 23
3. If f(x) = x^2 + 1 and g(x) = 5x, find (g o f)(4).
Show solution
- Core Practice: First identify exactly what the question is asking: If f(x) = x^2 + 1 and g(x) = 5x, find (g o f)(4).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- First compute f(4) = 17.
- Then compute g(17) = 5(17).
- The value is 85.
- Check the result by substituting or estimating: the response should match 85 and make sense in the original problem.
Answer: 85
4. If f(x) = x^2 + 1 and g(x) = 2x, find (g o f)(5).
Show solution
- Core Practice: First identify exactly what the question is asking: If f(x) = x^2 + 1 and g(x) = 2x, find (g o f)(5).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- First compute f(5) = 26.
- Then compute g(26) = 2(26).
- The value is 52.
- Check the result by substituting or estimating: the response should match 52 and make sense in the original problem.
Answer: 52
5. In (f o g)(x), which function acts first?
Choices: g · f · both at the same time · neither function
Show solution
- Core Practice: First identify exactly what the question is asking: In (f o g)(x), which function acts first?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Composition works inside out.
- g(x) becomes the input to f.
- So g acts first.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: g
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