CMClearMathAcademy

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

  1. Worked Example: First identify exactly what the question is asking: If f(x) = x + 3 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. Find f(3) = 3 + 3 = 6.
  4. Find g(3) = 2(3) = 6.
  5. Add the outputs: 6 + 6 = 12.
  6. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

Practice problems

1. If f(x) = x + 3 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 + 3 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. Find f(3) = 3 + 3 = 6.
  4. Find g(3) = 2(3) = 6.
  5. Add the outputs: 6 + 6 = 12.
  6. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = 3x + 4 and g(x) = x, find (f - g)(4).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Find f(4) = 3(4) + 4 = 16.
  4. Find g(4) = 4.
  5. Subtract: 16 - 4 = 12.
  6. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = x + 5 and g(x) = 1x + 5, find (g - f)(5).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Find g(5) = 1(5) + 5 = 10.
  4. Find f(5) = 5 + 5 = 10.
  5. Subtract in the order g - f: 10 - 10 = 0.
  6. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

4. If f(x) = x + 2 and g(x) = 2, find (fg)(6).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = x + 2 and g(x) = 2, find (fg)(6).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Find f(6) = 6 + 2 = 8.
  4. Find g(6) = 2.
  5. Multiply the outputs: 8 × 2 = 16.
  6. Check the result by substituting or estimating: the response should match 16 and make sense in the original problem.

Answer: 16

5. If f(x) = 3x and g(x) = x, find (f/g)(4).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = 3x and g(x) = x, find (f/g)(4).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Find f(4) = 3(4) = 12.
  4. Find g(4) = 4, which is not zero.
  5. Divide: 12 / 4 = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

Practice this interactively with instant feedback and an AI tutor.

Practice Function Operations Take the free placement check

More Precalculus lessons