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Function Notation

A free College Algebra lesson from the “Functions” unit, with a worked example and practice problems including step-by-step solutions.

Function notation names a rule. If f(x) = 2x + 3, then f(4) means use 4 as the input. The output is found by substituting the input into the expression.

What you'll learn

Why it matters: Calculators, spreadsheets, tax tables, and science formulas all use function notation to show which input is being used. Reading f(5) as the output for input 5 helps prevent the common mistake of treating notation like multiplication.

Worked example

Problem. If f(x) = 3x - 5, find f(6).

  1. Replace x with 6.
  2. Compute 3(6) - 5.
  3. 18 - 5 = 13.

Answer: 13

Practice problems

1. If g(x) = x^2 + 1, find g(5).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If g(x) = x^2 + 1, find g(5).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Replace x with 5.
  4. 5^2 + 1 = 25 + 1 = 26.
  5. Check the result by substituting or estimating: the response should match 26 and make sense in the original problem.

Answer: 26

2. If h(x) = 4x - 9, what does h(2) mean?

Choices: Multiply h by 2 · Use 2 as the input · Set h equal to 2 · Divide by 2

Show solution
  1. Core Practice: First identify exactly what the question is asking: If h(x) = 4x - 9, what does h(2) mean?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The number in parentheses is the input value.
  4. h(2) means evaluate h when x = 2.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Use 2 as the input

3. If p(x) = 2x^2 - 3, find p(4).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If p(x) = 2x^2 - 3, find p(4).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. 2(16) - 3 = 29.
  4. Check the result by substituting or estimating: the response should match 29 and make sense in the original problem.

Answer: 29

4. If f(x) = x^2 - 2x, find f(5).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = x^2 - 2x, find f(5).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute 5: 25 - 10 = 15.
  4. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

5. If f(3) = 12, then 12 is the...

Choices: Output when the input is 3 · Input when the output is 3 · Slope of f · Name of the function

Show solution
  1. Interpretation: First identify exactly what the question is asking: If f(3) = 12, then 12 is the...
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The value in parentheses is the input.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Output when the input is 3

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