Video Transcript
Which of these sets of ordered pairs does not represent a function? Set (a) the set containing one, two; two, four; three, six; four, eight. Set (b) the set containing one, two; two, four; three, six; and three, eight. Set (c) the set containing two, 10; three, 15; four, 20; five, 25. And set (d) the set containing two, 10; two, 15; four, 20; four, 25. Is it option (A) b and d, option (B) a and d, option (C) b and c, option (D) b? Or is it option (E) c?
In this question, we’re given four sets of ordered pairs. And we need to determine which of these sets does not represent a function. And to help us determine this, let’s start by recalling what we mean by a function.
We recall a function takes every member of a set and maps it onto exactly one member in the second set. And the most important part of this definition is that the inputs are mapped onto exactly one member of the outputs. This means if we input a number into a function, we will always get the same output value.
However, in this question, we’re not given functions. We’re given sets of ordered pairs. So let’s also recall how an ordered pair can represent a function. First, when we’re talking about ordered pairs representing a function, the first value is the input value and the second value is the corresponding output value. So if the ordered pair 𝑥, 𝑦 was an element of a relation which represented a function, then 𝑥 would be the input value and 𝑦 would be the corresponding output value. But remember, for a relation to represent a function, every member of the set can only be mapped onto exactly one member of the output set. And we can ask the question, what does this mean for our relation?
Well, if each input value needs to be mapped onto exactly one output value, then the input value can only appear once in the set. For example, if we look at option (b), we can see that the ordered pair three, six and the ordered pair three, eight lies in the set. And in particular, we need to notice the input value of three would appear twice. This would mean when we input a value of three, we need to output a value of six. And when we input a value of three, we need to output a value of eight. Since there’s two possible output values for three, this can’t represent a function. So (b) does not represent a function. And remember, we’re looking for the sets of ordered pairs which do not represent a function. So (b) is one of the sets we want to take.
We can also see that something very similar is true in set (d). There are two ordered pairs with first element four. So if we wanted this to represent a function, four would need to be mapped to 20 and four would also need to be mapped to 25. Therefore, four is not mapped onto exactly one member of the range. So this can’t represent a function either. And this is enough to conclude that the answer is option (A) b and d do not represent functions.
However, for due diligence, let’s also check that (a) and (c) do represent functions. Let’s start with (a). For this set of ordered pairs to represent a function, the input values need to be mapped to one element of the outputs. In other words, the first entry in the ordered pairs needs to only appear once in the set. We can see this is true. One is mapped to two, two is mapped to four, three is mapped to six, and four is mapped to eight. And each of our input values only appears once. So (a) represents a function. The exact same is true in option (c). Two, three, four, and five are the input values, and they only appear once. So (c) also represents a function.
Therefore, we were able to show, of the four given sets of ordered pairs, only options (b) and (d) did not represent a function.
