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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

Why it matters: Discount-then-tax checkout totals, scaled-and-shifted graph designs, and loans with stacked fees are composed functions. The output of the first rule becomes the input of the second, so order matters — f(g(x)) and g(f(x)) usually give different answers.

Worked example

Problem. If f(x) = 2x + 1 and g(x) = x - 4, find f(g(10)).

  1. Evaluate inside first: g(10) = 6.
  2. Now find f(6).
  3. 2(6) + 1 = 13.

Answer: 13

Practice problems

1. If f(x) = x + 3 and g(x) = 2x, find f(g(4)).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = x + 3 and g(x) = 2x, find f(g(4)).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. g(4) = 8.
  4. f(8) = 11.
  5. Check the result by substituting or estimating: the response should match 11 and make sense in the original problem.

Answer: 11

2. If f(x) = 5x and g(x) = x - 1, find g(f(2)).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = 5x and g(x) = x - 1, find g(f(2)).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(2) = 10.
  4. g(10) = 9.
  5. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

3. If f(x) = x^2 and g(x) = x + 2, find f(g(3)).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = x^2 and g(x) = x + 2, find f(g(3)).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. g(3) = 5.
  4. f(5) = 25.
  5. Check the result by substituting or estimating: the response should match 25 and make sense in the original problem.

Answer: 25

4. If f(x) = 3x - 7 and g(x) = x^2, find f(g(4)).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = 3x - 7 and g(x) = x^2, find f(g(4)).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. g(4) = 16.
  4. 3(16) - 7 = 41.
  5. Check the result by substituting or estimating: the response should match 41 and make sense in the original problem.

Answer: 41

5. For f(g(x)), which function happens first?

Choices: g · f · Both at once · Neither

Show solution
  1. Challenge: First identify exactly what the question is asking: For f(g(x)), which function happens first?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The inside function is evaluated first.
  4. 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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