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Function Operations

A free Precalculus lesson from the “Transformations and Combinations of Functions” unit, with a worked example and practice problems including step-by-step solutions.

Function operations combine outputs, but division adds domain restrictions wherever the denominator function is zero. 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) = x + 2 and g(x) = 2x, find (f + g)(3).

  1. Worked Example: First identify exactly what the question is asking: If f(x) = x + 2 and g(x) = 2x, find (f + g)(3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(3) = 5.
  4. g(3) = 6.
  5. Add outputs: 5 + 6 = 11.
  6. Check the result by substituting or estimating: the response should match 11 and make sense in the original problem.

Answer: 11

Practice problems

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = x + 2 and g(x) = 2x, find (f + g)(3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(3) = 5.
  4. g(3) = 6.
  5. Add outputs: 5 + 6 = 11.
  6. 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) = x + 3 and g(x) = 3x, 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) = 3x, find (f + g)(4).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(4) = 7.
  4. g(4) = 12.
  5. Add outputs: 7 + 12 = 19.
  6. Check the result by substituting or estimating: the response should match 19 and make sense in the original problem.

Answer: 19

3. If f(x) = x^2 and g(x) = x - 4, find (fg)(5).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = x^2 and g(x) = x - 4, find (fg)(5).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(5) = 25.
  4. g(5) = 1.
  5. Multiply outputs: 25(1) = 25.
  6. 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) = x^2 and g(x) = x - 5, find (fg)(2).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = x^2 and g(x) = x - 5, find (fg)(2).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(2) = 4.
  4. g(2) = -3.
  5. Multiply outputs: 4(-3) = -12.
  6. Check the result by substituting or estimating: the response should match -12 and make sense in the original problem.

Answer: -12

5. For (f/g)(x), if g(x) = x - 1, what value is excluded?

Show solution
  1. Core Practice: First identify exactly what the question is asking: For (f/g)(x), if g(x) = x - 1, what value is excluded?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The denominator function cannot be zero.
  4. Set x - 1 = 0.
  5. The excluded value is 1.
  6. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

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