True or False: The following set of ordered pairs represents a function: the set containing one, three; two, six; three, nine; four, 12.
In this question, we’re given a set of ordered pairs, and we need to determine if this set represents a function. So to do this, let’s start by recalling what we mean by a function. These map every element of one set onto exactly one member of a second set. And there’s a few important things to note. For example, the two sets mentioned can be equal. And the important thing to notice they map the inputs onto exactly one member in the output set. And this means when we input something into a function, we always get the same output.
However, in this question, we’re not given something in function notation; we’re given a set of ordered pairs. And a set of ordered pairs is called a relation. In fact, it’s a binary relation. And so to answer this question, we need to recall how a relation can represent a function. We recall for a relation to represent a function, the first value of the ordered pair represents the input value and the second value, the output value. For example, if 𝑥, 𝑦 is an element of our relation, then we want to say that 𝑥 is the input value of the function and 𝑦 is the corresponding output value.
We could now start looking at the relation given to us in the question. However, we could also ask the question, how do we guarantee that a relation represents a function? We know that functions map every element of one set onto exactly one member in the second set. In other words, the inputs are mapped to exactly one output. And we can ask the question, what does this mean for our relation? If the first value of our ordered pairs represents the input values and we need these to be mapped onto exactly one member of the output set, then we need each input value to only appear once in our relation. Otherwise, we’d be trying to map input values onto two different output values.
And we can see this is true in this set. The input value of one only appears once, two only appears once, three only appears once, and four only appears once. We don’t even need to look at the output values to determine whether this relation represents a function. Therefore, we were able to show it’s true that the relation the set containing one, three; two, six; three, nine; four, 12 represents a function.