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Functions and Notation

A free Algebra I lesson from the “Functions, Linear Relationships, and Rate of Change” unit, with a worked example and practice problems including step-by-step solutions.

A function is a rule that gives exactly one output for each input. Function notation, such as f(3), means substitute 3 into the rule named f and simplify. This matters because functions let us describe patterns, tables, graphs, and real-world relationships with one consistent language. When practicing, first identify the input, then replace the variable with that input, and finally simplify in the correct order. A common mistake is to treat f(x) as multiplication by f; it is not. It is the name of the rule and the input being used.

What you'll learn

Why it matters: Apps, spreadsheets, sensors, and game rules use functions to turn an input into a predictable output.

Worked example

Problem. If f(x) = 2x - 1, find f(6).

  1. Replace x with 6.
  2. Compute 2(6) - 1.
  3. 12 - 1 = 11.

Answer: 11

Practice problems

1. If f(x) = x + 4, find f(9).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = x + 4, find f(9).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute 9 for x.
  4. 9 + 4 = 13.
  5. Check the result by substituting or estimating: the response should match 13 and make sense in the original problem.

Answer: 13

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

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

Answer: 15

3. In the ordered pair (2, 7), what is the input?

Choices: 2 · 7 · 9 · 5

Show solution
  1. Warm-up: First identify exactly what the question is asking: In the ordered pair (2, 7), what is the input?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The input is the x-value.
  4. In (2, 7), the input is 2.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2

4. If h(x) = 4x - 3, find h(6).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If h(x) = 4x - 3, find h(6).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute 6.
  4. 4(6) - 3 = 24 - 3 = 21.
  5. Check the result by substituting or estimating: the response should match 21 and make sense in the original problem.

Answer: 21

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = x^2 + 2, find f(4).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute 4.
  4. 4^2 + 2 = 16 + 2 = 18.
  5. Check the result by substituting or estimating: the response should match 18 and make sense in the original problem.

Answer: 18

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