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
- Decide whether a relation is a function and explain why
- Use what a function is in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. Which relation is a function?
- A function pairs each input with exactly one output.
- In {(2, 3), (3, 4), (4, 5)} no input repeats, so every input has a single output.
- Repeated outputs are fine; repeated inputs with different outputs are not.
Answer: {(2, 3), (3, 4), (4, 5)}
Practice problems
1. Which relation is a function?
Choices: {(2, 3), (3, 4), (4, 5)} · {(2, 3), (2, 4), (4, 5)} · {(3, 3), (3, 5), (4, 4)} · {(4, 3), (4, 4), (2, 5)}
Show solution
- A function pairs each input with exactly one output.
- In {(2, 3), (3, 4), (4, 5)} no input repeats, so every input has a single output.
- Repeated outputs are fine; repeated inputs with different outputs are not.
Answer: {(2, 3), (3, 4), (4, 5)}
2. Why is {(3, 4), (3, 5), (5, 6)} not a function?
Choices: the input 3 has two different outputs · one of the outputs is negative · the inputs are too large · it has three ordered pairs
Show solution
- Scan for a repeated input in {(3, 4), (3, 5), (5, 6)}.
- The input 3 is paired with 4 and 5.
- One input with two outputs breaks the function rule.
Answer: the input 3 has two different outputs
3. In the relation {(4, 5), (4, 6), (6, 7)}, which input value is paired with two different outputs?
Show solution
- Core Practice: First identify exactly what the question is asking: In the relation {(4, 5), (4, 6), (6, 7)}, which input value is paired with two different outputs?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Look for the x-value that appears more than once.
- The pair 4 shows up with 5 and with 6.
- So the repeated input is 4.
- Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.
Answer: 4
4. In a function machine, one input must produce:
Choices: exactly one output · every possible output · at least two outputs · a larger number than the input
Show solution
- A function machine is deterministic.
- Feed in one input and a single output comes out.
- Exactly one output is the defining condition.
Answer: exactly one output
5. A graph represents a function when every vertical line crosses it:
Choices: at most once · at least twice · exactly twice · never
Show solution
- This is the vertical line test.
- Each input (x) may have only one output (y).
- If a vertical line hits twice, that x has two y-values.
Answer: at most once
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