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What a Function Is

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

A function assigns each input exactly one output. The input can repeat only if the output repeats with it. 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. Which relation is a function?

  1. A function assigns each input exactly one output.
  2. In {(-2, 7), (0, 7), (4, 1)}, no input is paired with two different outputs.
  3. Repeated outputs do not break the function rule.

Answer: {(-2, 7), (0, 7), (4, 1)}

Practice problems

1. Which relation is a function?

Choices: {(-2, 7), (0, 7), (4, 1)} · {(-2, 7), (0, 1), (-2, 3)} · {(2, 6), (2, 8), (9, 1)} · {(0, 0), (0, 2), (5, 5)}

Show solution
  1. A function assigns each input exactly one output.
  2. In {(-2, 7), (0, 7), (4, 1)}, no input is paired with two different outputs.
  3. Repeated outputs do not break the function rule.

Answer: {(-2, 7), (0, 7), (4, 1)}

2. Which relation is a function?

Choices: {(3, -1), (4, -1), (5, 8)} · {(3, -1), (4, 0), (3, 8)} · {(2, 6), (2, 8), (9, 1)} · {(0, 0), (0, 2), (5, 5)}

Show solution
  1. A function assigns each input exactly one output.
  2. In {(3, -1), (4, -1), (5, 8)}, no input is paired with two different outputs.
  3. Repeated outputs do not break the function rule.

Answer: {(3, -1), (4, -1), (5, 8)}

3. Why is {(1, 2), (1, 4), (3, 5)} not a function?

Choices: input 1 has two different outputs · one output is negative · the x-values are too large · the relation has ordered pairs

Show solution
  1. Look for a repeated input in {(1, 2), (1, 4), (3, 5)}.
  2. The input 1 appears with different outputs.
  3. That violates the function rule.

Answer: input 1 has two different outputs

4. Why is {(-2, 7), (0, 1), (-2, 3)} not a function?

Choices: input -2 has two different outputs · one output is negative · the x-values are too large · the relation has ordered pairs

Show solution
  1. Look for a repeated input in {(-2, 7), (0, 1), (-2, 3)}.
  2. The input -2 appears with different outputs.
  3. That violates the function rule.

Answer: input -2 has two different outputs

5. In a function machine, what must be true about one input?

Choices: it produces exactly one output · it must produce every output · it must be positive · it must be larger than the output

Show solution
  1. A function can have many possible inputs.
  2. For any one input, the output must be unambiguous.
  3. Exactly one output is the key condition.

Answer: it produces exactly one output

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