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Function Notation

A free Precalculus lesson from the “Function Foundations and Behavior” unit, with a worked example and practice problems including step-by-step solutions.

Function notation is input-output language. The value inside parentheses is the input, not something to multiply. 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: Function behavior is the language behind Calculus, physics graphs, economic models, and computer-generated curves.

Worked example

Problem. If f(x) = 3x + 2, find f(3).

  1. Worked Example: First identify exactly what the question is asking: If f(x) = 3x + 2, find f(3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute x = 3 into the rule.
  4. 3(3) + 2 = 9 + 2.
  5. So f(3) = 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) = 3x + 2, find f(3).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = 3x + 2, find f(3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute x = 3 into the rule.
  4. 3(3) + 2 = 9 + 2.
  5. So f(3) = 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) = 4x - 3, find f(-3).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = 4x - 3, find f(-3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute x = -3.
  4. 4(-3) - 3 = -12 - 3.
  5. So f(-3) = -15.
  6. Check the result by substituting or estimating: the response should match -15 and make sense in the original problem.

Answer: -15

3. If g(x) = x^2 + 4, find g(2).

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

Answer: 8

4. If g(x) = x^2 - 2, find g(-1).

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

Answer: -1

5. If f(x) = x^3, find f(4).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = x^3, find f(4).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute x = 4.
  4. 4^3 = 4 · 4 · 4 = 64.
  5. So f(4) = 64.
  6. Check the result by substituting or estimating: the response should match 64 and make sense in the original problem.

Answer: 64

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