The mapping diagram represents a relation where x represents the independent variable and y represents the dependent variable.

Question

A mapping diagram with one circle labeled x-values containing values negative 4, negative 2, 0, 1, and 3 and another circle labeled y values containing values negative 5, negative 4, negative 3, negative 2, and negative 1 and arrows from negative 4 to negative 5, negative 2 to negative 3, 0 to negative 4, 0 to negative 2, 1 to negative 3, and 3 to negative 1.

Is the relation a function? Explain.
No, because for each input there is not exactly one output
No, because for each output there is not exactly one input
Yes, because for each input there is exactly one output
Yes, because for each output there is exactly one input

Answer 1

To determine whether the relation represented by the mapping diagram is a function, we need to recall the definition of a function: a relation is a function if every input (x-value) is associated with exactly one output (y-value).

In the mapping you described, we can analyze the relationships:

• \( x = -4 \) maps to \( y = -5 \) (one output)
• \( x = -2 \) maps to \( y = -3 \) (one output)
• \( x = 0 \) maps to both \( y = -4 \) and \( y = -2 \) (two outputs)
• \( x = 1 \) maps to \( y = -3 \) (one output)
• \( x = 3 \) maps to \( y = -1 \) (one output)

The crucial point is that the x-value \( 0 \) is associated with two different y-values (\( -4 \) and \( -2 \)). This means that for the input \( 0 \), there is not exactly one output.

Therefore, the relation is not a function because for the input \( 0 \), there are two outputs.

The correct explanation would be: No, because for each input there is not exactly one output.