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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 - 1, find f(-2).

  1. Worked Example: First identify exactly what the question is asking: If f(x) = 3x - 1, find f(-2).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute -2 for x.
  4. Compute 3(-2) - 1.
  5. The output is -7.
  6. Check the result by substituting or estimating: the response should match -7 and make sense in the original problem.

Answer: -7

Practice problems

1. If f(x) = 3x - 1, find f(-2).

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

Answer: -7

2. If f(x) = 4x + 0, find f(-1).

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

Answer: -4

3. If g(x) = x^2 + 5, find g(0).

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

Answer: 5

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 1 for x.
  4. Square first: 1^2 = 1.
  5. Add 2 to get 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

5. If h(t) = 3t - 1 and h(t) = 5, find t.

Show solution
  1. Core Practice: First identify exactly what the question is asking: If h(t) = 3t - 1 and h(t) = 5, find t.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set 3t - 1 = 5.
  4. Add 1 to both sides to get 3t = 6.
  5. Divide by the coefficient to get t = 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

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