Question

does this table represent a function?
input 2 4 4 6
output 4 5 5 6

Explanation:

Step1: Recall the definition of a function

A function is a relation where each input (or domain value) is associated with exactly one output (or range value). In other words, for a relation to be a function, no input value can be mapped to more than one output value.

Step2: Analyze the given table

Looking at the input values:
- The input value \( 2 \) is mapped to the output value \( 4 \) (only one output).
- The input value \( 4 \) is mapped to the output value \( 5 \) (and it is mapped to \( 5 \) again, but it's still the same output for the same input; having the same input mapped to the same output multiple times is allowed).
- The input value \( 6 \) is mapped to the output value \( 6 \) (only one output).

Wait, hold on. Wait, the key point is: does any input have more than one different output? Let's check again. The input \( 4 \) is mapped to \( 5 \) in both cases where the input is \( 4 \). So the input \( 4 \) has only one output (which is \( 5 \)), even though it appears twice as an input. The input \( 2 \) has output \( 4 \), input \( 4 \) has output \( 5 \), input \( 6 \) has output \( 6 \). Wait, but wait, the definition is that each input must have exactly one output. So if an input is repeated, but it maps to the same output, that's okay. For example, if input \( 4 \) was mapped to \( 5 \) and \( 6 \), then it would not be a function. But in this case, input \( 4 \) is mapped to \( 5 \) both times. So each input has exactly one output. Wait, but wait, let's list the ordered pairs: \( (2,4) \), \( (4,5) \), \( (4,5) \), \( (6,6) \). In a function, a relation is a set of ordered pairs where no two ordered pairs have the same first element (input) with different second elements (output). Here, the input \( 4 \) is paired with \( 5 \) in both cases, so the first elements are the same, but the second elements are the same. So there's no input that has different outputs. So does this table represent a function?

Wait, maybe I made a mistake earlier. Let's re-express the definition: A function is a relation from a set \( A \) to a set \( B \) such that each element in \( A \) is assigned to exactly one element in \( B \). So if an element in \( A \) (input) is assigned to more than one element in \( B \) (output), then it's not a function. In the given table, the input \( 4 \) is assigned to \( 5 \) both times. So it's assigned to only one element in \( B \) (which is \( 5 \)). So the input \( 4 \) has only one output. The input \( 2 \) has output \( 4 \), input \( 6 \) has output \( 6 \). So all inputs have exactly one output. Wait, but wait, the table has two entries with input \( 4 \), but both have output \( 5 \). So the relation is a function?

Wait, no, wait. Wait, the problem is: does the table represent a function? Let's check the definition again. A function is a set of ordered pairs where no two ordered pairs have the same first component (input) with different second components (output). So in this case, the ordered pairs are \( (2,4) \), \( (4,5) \), \( (4,5) \), \( (6,6) \). The first components are \( 2 \), \( 4 \), \( 4 \), \( 6 \). The second components are \( 4 \), \( 5 \), \( 5 \), \( 6 \). Now, the first component \( 4 \) appears twice, but both times with the same second component \( 5 \). So there are no two ordered pairs with the same first component and different second components. Therefore, this relation is a function?

Wait, but maybe I'm confused. Let's take an example where it's not a function. Suppose we have input \( 4 \) mapped to \( 5 \) and \( 6 \). Then, the ordered pairs would be \( (4,5) \) and \( (4,6) \), which have the same first component and different second components, so that's not a function. But in our problem, the input \( 4 \) is mapped to \( 5 \) both times, so the second components are the same. Therefore, it is a function?

Wait, but let's check the original table again. The input values are \( 2 \), \( 4 \), \( 4 \), \( 6 \). The output values are \( 4 \), \( 5 \), \( 5 \), \( 6 \). So for each input, there's only one output. The input \( 4 \) has output \( 5 \) (even though it's listed twice, it's the same output). So yes, it is a function?

Wait, no, wait. Wait, the definition of a function is that each input has exactly one output. So if an input is repeated, but it always gives the same output, that's okay. For example, if we have a function \( f(x) = x + 2 \), and we plug in \( x = 4 \) twice, we get \( 6 \) both times. So the relation is still a function. So in this table, even though \( 4 \) is an input twice, it always gives \( 5 \) as the output. So each input has exactly one output. Therefore, the table represents a function.

Wait, but maybe I made a mistake. Let me check again. The key is: does any input have more than one different output? Let's list the inputs and their corresponding outputs:

- Input \( 2 \): Output \( 4 \) (only one output)
- Input \( 4 \): Output \( 5 \) (only one output, even though it's repeated)
- Input \( 6 \): Output \( 6 \) (only one output)

So since each input has exactly one output, the table represents a function.

Answer:

Yes, the table represents a function because each input value is associated with exactly one output value (the input 4 is associated with output 5 in both cases, so it has only one unique output, and other inputs also have unique outputs).