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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

Why it matters: Transformations let students predict how a model changes when a situation is shifted, scaled, reflected, or combined.

Worked example

Problem. If f(x) = 3x + 2 and g(x) = x + 2, find (f o g)(3). Work inside out.

  1. Worked Example: First identify exactly what the question is asking: If f(x) = 3x + 2 and g(x) = x + 2, find (f o g)(3). Work inside out.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. First the inner function: g(3) = 3 + 2 = 5.
  4. Feed that into f: f(5) = 3(5) + 2.
  5. 3(5) + 2 = 15 + 2 = 17.
  6. Check the result by substituting or estimating: the response should match 17 and make sense in the original problem.

Answer: 17

Practice problems

1. If f(x) = 3x + 2 and g(x) = x + 2, find (f o g)(3). Work inside out.

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = 3x + 2 and g(x) = x + 2, find (f o g)(3). Work inside out.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. First the inner function: g(3) = 3 + 2 = 5.
  4. Feed that into f: f(5) = 3(5) + 2.
  5. 3(5) + 2 = 15 + 2 = 17.
  6. Check the result by substituting or estimating: the response should match 17 and make sense in the original problem.

Answer: 17

2. If f(x) = 4x + 3 and g(x) = x + 3, find (g o f)(4). Work inside out.

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = 4x + 3 and g(x) = x + 3, find (g o f)(4). Work inside out.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. First the inner function: f(4) = 4(4) + 3 = 19.
  4. Feed that into g: g(19) = 19 + 3.
  5. 19 + 3 = 22.
  6. Check the result by substituting or estimating: the response should match 22 and make sense in the original problem.

Answer: 22

3. Let f(x) = 2x + 4 and g(x) = x + 1. Compute (f o g)(5) - (g o f)(5) to see that order matters.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Let f(x) = 2x + 4 and g(x) = x + 1. Compute (f o g)(5) - (g o f)(5) to see that order matters.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. (f o g)(5) = 16 and (g o f)(5) = 15.
  4. Subtract: 16 - 15.
  5. The difference is 1, so the two orders disagree.
  6. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

4. If f(x) = 3x + 1 and g(x) = x + 2, write f(g(x)) as a simplified expression of the form (slope)x + (number). Enter the number (the constant term).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = 3x + 1 and g(x) = x + 2, write f(g(x)) as a simplified expression of the form (slope)x + (number). Enter the number (the constant term).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Substitute g into f: f(g(x)) = 3(x + 2) + 1.
  4. Distribute: 3x + 6 + 1.
  5. The constant term is 6 + 1 = 7.
  6. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

5. If f(x) = 4x + 2 and g(x) = x + 3, write g(f(x)) as a simplified expression of the form (slope)x + (number). Enter the number (the constant term).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = 4x + 2 and g(x) = x + 3, write g(f(x)) as a simplified expression of the form (slope)x + (number). Enter the number (the constant term).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Substitute f into g: g(f(x)) = (4x + 2) + 3.
  4. Combine constants: 4x + 2 + 3.
  5. The constant term is 2 + 3 = 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

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