### Video Transcript

Which of the following relations represents a function?

In this question, we are presented with three mapping diagrams. These diagrams are used to represent the relationship between input values and output values. We recall that a relation, also called a mapping, transforms elements from one set to another. We notice that for each of the three mapping diagrams provided, the set of inputs includes 𝑎, 𝑏, and 𝑐 and the set of outputs includes one, two, three, and four.

We recall that not all relations are functions. In fact, a function is a special type of relation. A function is a relation where every input value maps to a single output value. We will begin by looking at diagram (A). We notice that elements 𝑎 and 𝑐 map onto the single elements one and three, respectively. These inputs do have exactly one output each. However, we also know that for a mapping diagram to represent a function, every input value must have an output. Element 𝑏 does not map onto any element in the second set. Since element 𝑏 has no output, this mapping diagram does not represent a function. In fact, this is an example of an invalid mapping diagram, since a mapping diagram should provide an output for all inputs.

Next, we consider diagram (B). In this mapping diagram, every input has an output. However, the element 𝑐 is mapped on to two and three in the second set. For a mapping diagram to represent a function, every input value must have a single output. Yet element 𝑐 maps to two different outputs. This means that the relation in diagram (B) cannot represent a function. We might deduce that the final mapping diagram (C) represents a function, but we will need to verify this by checking that each input has exactly one output. We will do this by checking the mapping of each input value. 𝑎 maps onto three and nothing else, 𝑏 maps onto one and nothing else, and 𝑐 maps onto three and nothing else.

We have now verified that every input maps onto a single output. We should note that each input does not have to map onto a unique output. This means that it is okay that 𝑎 maps onto three and 𝑐 also maps onto three. Two inputs can map onto the same output as long as one input does not map onto two different outputs. Therefore, of the three mapping diagrams given, only diagram (C) represents a function.