To determine if the relation shown in the table is a function, we need to check if every input \( x \) has exactly one unique output \( f(x) \).
Looking at the table:
- For \( x = 3 \), \( f(x) = 2 \)
- For \( x = 0 \), \( f(x) = 1 \)
- For \( x = 5 \), \( f(x) = -7 \)
- For \( x = -3 \), \( f(x) = 2 \)
In this case, the input \( x \) values (3, 0, 5, -3) are all unique. Each \( x \) maps to a single unique \( f(x) \). However, notice that \( f(x) = 2 \) is obtained when \( x = 3 \) and also when \( x = -3 \). This means that multiple inputs have the same output, but each input still has only one output.
Now let's analyze the responses given:
- Yes, every input has one unique output. - Correct.
- Yes, every output has one unique input. - Incorrect, since both \( x = 3 \) and \( x = -3 \) yield the same output (2).
- No, there are multiple outputs with the same input. - Incorrect, because each \( x \) has only one corresponding \( f(x) \).
- No, there are multiple inputs with the same output. - Correct in a way, but not relevant for defining a function.
So, the best response here is:
Yes, every input has one unique output.
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