Question

Is the relation shown in the table a function? Why or why not?
x | f(x)
3 | 2
0 | 1
5 | -7
-3 | 2
A. No, there are multiple outputs with the same input.
B. No, there are multiple inputs with the same output.
C. Yes, every output has one unique input.
D. Yes, every input has one unique output.

Answer 1

To determine if the relation shown in the table is a function, we need to check whether every input (x-value) corresponds to exactly one output (f(x)).

In the provided table:

- When \( x = 3 \), \( f(x) = 2 \)
- When \( x = 0 \), \( f(x) = 1 \)
- When \( x = 5 \), \( f(x) = -7 \)
- When \( x = -3 \), \( f(x) = 2 \)

Here, the input \( 3 \) maps to \( 2 \), \( 0 \) maps to \( 1 \), \( 5 \) maps to \( -7 \), and \( -3 \) also maps to \( 2 \).

The key observation is that while the input \( -3 \) yields the same output \( 2 \) as the input \( 3 \), each input still has one unique output. Therefore, there are no repeated inputs producing different outputs, and thus, there are no violations of the definition of a function.

Based on this reasoning, the correct answer is:

**D. Yes, every input has one unique output.**