Questions: Relations and Functions

Question 1

Which relation is a function?

• A. {(1, 2), (1, 3), (2, 4)}
• B. {(1, 5), (2, 5), (3, 5)}
• C. {(0, 0), (4, 2), (4, -2)}
• D. {(3, 1), (3, 1), (3, 2)}

Answer: (B)

In {(1, 5), (2, 5), (3, 5)} each input 1, 2, 3 has exactly one output. All outputs being 5 is allowed. The other options each pair one input with two different outputs: 1 with 2 and 3, 4 with 2 and -2, and 3 with 1 and 2.

Question 2

A relation in which two different inputs have the same output cannot be a function.

• T. True
• F. False

Answer: False

The definition restricts inputs, not outputs. Each input must have exactly one output, but different inputs may share an output. For example, every input of f(x) = 7 has output 7.

Question 3

Does the graph of the circle x² + y² = 25 represent y as a function of x? Explain using the vertical line test.

Model answer: No. The vertical line x = 3 meets the circle at (3, 4) and (3, -4), so the input 3 has two outputs.

A vertical line that crosses a graph twice shows one x-value paired with two y-values. Most vertical lines through the interior of this circle cross it twice, so the circle fails the test.

Question 4

A table lists inputs 1, 2, 3, 2 with outputs 4, 6, 8, 6. Is this a function?

• A. No, because the input 2 appears twice
• B. Yes, because the input 2 is paired with the same output both times
• C. No, because the output 6 appears twice
• D. Yes, because there are four inputs and four outputs

Answer: (B)

A repeated row such as (2, 6) listed twice is the same pair, so 2 still has only one output. The relation would fail only if 2 were paired with two different outputs.