Question

which statement describes how to determine if a relation given in a table is a function?

if none of the output values are repeated, the relation is a function.
if none of the input values are repeated, the relation is a function.
if any of the input values are equal to the output values, the relation is a function.
if all the output values are paired with the same input value, the relation is a function.

Explanation:

Define a mathematical function

Using the Function Definition knowledge point
A relation is a function if and only if each input value (domain) maps to exactly one output value (range). This means that for any given input value, there cannot be multiple different output values associated with it.

Analyze the role of input values

Using the Function Definition knowledge point
If any input value is repeated in a table, we must check if it pairs with different output values. If no input values are repeated, then each unique input automatically pairs with exactly one output, guaranteeing that the relation is a function.

Evaluate the given choices

Using the Function Definition knowledge point

• Option 1: "If none of the output values are repeated..." — Incorrect. Output values can repeat (e.g., \(y = c\)).
• Option 2: "If none of the input values are repeated..." — Correct. If every input is unique, each maps to exactly one output.
• Option 3: "If any of the input values are equal to the output values..." — Incorrect. This is not a requirement for a function.
• Option 4: "If all the output values are paired with the same input value..." — Incorrect. This describes a relation that is not a function if there are multiple outputs for that single input.

Answer:

• If none of the output values are repeated, the relation is a function.
• If none of the input values are repeated, the relation is a function. (Correct answer)
• If any of the input values are equal to the output values, the relation is a function.
• If all the output values are paired with the same input value, the relation is a function.