Lesson Video: Relations and Functions Mathematics • Third Year of Preparatory School

Video Transcript

In this video, we will learn how to identify, represent, and recognize functions from schematic descriptions, arrow diagrams, and graphs. We will begin the video by describing what we mean by relations and functions.

A relation or relationship describes a property between the objects of two sets. If the object 𝑎 sub three from the first set exhibits this relationship to the object 𝑏 sub seven in the second set, then it can be represented with a relation diagram. We draw an arrow from 𝑎 sub three to 𝑏 sub seven. Let’s now consider a real-life example.

The following diagram represents the relationship “son or daughter of.” The first set of names Emma, Chloe, Jennifer, Noah, and Scarlett are the sons and daughters. The second set of names are the mothers or fathers. If we consider Emma, we can see that Emma is the daughter of Madison and Liam. We can, therefore, deduce that Chloe and Emma are sisters as Madison and Liam are the mother and father to both girls. Jennifer and Noah both have the same mother, Amelia, but they have different fathers. Antony is Jennifer’s father, and Daniel is Noah’s father.

We will now define what we mean by a function. A function is a special type of relation that connects each object of the first set, called the input, to exactly one object of the second set, called the output. When we talk about functions, the objects are usually numbers. We will look at a real-life example first.

The relation “lives in” is a function since each person in the first set is associated with only one city in the second set. We can see from the diagram that Hannah lives in Detroit, David lives in New York, both Ethan and Victoria live in Boston, and none of the four people live in Chicago. This is clearly a function as each element of the first set is connected to exactly one object of the other set. Only one arrow goes from each object.

Relations can also be represented as a set of ordered pairs. In this case, we would have four ordered pairs: Hannah in Detroit, David in New York, Ethan in Boston, and Victoria in Boston. The first name in the ordered pair is the input and the second name, the city in this case, is the output. In order for our relation to be a function, each of the inputs must be unique.

We will now look at some questions involving relations and functions.

Determine whether the following statement is true or false: the given figure represents a function.

We have two sets of values. We have the 𝑥- or input values of four, five, and eight and the 𝑦-output values of two, five, seven, and nine. We are asked to decide whether this figure represents a function. We recall that a function is a relation where each object in the first set or input connects to exactly one object in the second set or output. Number four in the first set only connects to number seven in the second set. Likewise, the number eight in the first set or input connects to only number five in the second set or output.

So far, each object connects to exactly one object in the output. We have a problem, however, with the number five in the input as this connects to both two and nine. This means that the given figure does not represent a function and the correct answer is false. The figure is not a function as five connects to both two and nine.

In our next question, we need to determine which of the three diagrams represents a function.

Which of the following relations represents a function? Is it (A) 𝑎 to one and 𝑐 to three; option (B) 𝑎 to one, 𝑏 to two, 𝑐 to two, and 𝑐 to three; or option (C) 𝑎 to three, 𝑏 to one, and 𝑐 to three?

In order to answer this question, we need to recall our definition of a function. A function is a relation where each object in the first set or input is connected to exactly one object in the second set. This is known as the output. The keywords here are exactly one. In option (A), the letter 𝑏 in the input is not connected to any number in the output. This means that this relation does not represent a function.

In option (B), the letters 𝑎 and 𝑏 are connected to exactly one object in the second set; 𝑎 connects to one and 𝑏 connects to two. However, the letter 𝑐 is connected to two objects in the second set; it is connected to the number two and the number three. This means that relation (B) again does not represent a function.

In option (C), the letter 𝑎 is only connected to the number three. The letter 𝑏 is only connected to the number one. Finally, the letter 𝑐 is only connected to the number three. Each object in the first set, the letters 𝑎, 𝑏, and 𝑐, are connected to exactly one object in the second set. This means that the correct answer is (C). This is the only relation that represents a function.

Our next question involves identifying a function from a table.

Which of the following represents a function with input 𝑥 and output 𝑦? Both relation A and relation B have five 𝑥-input values and five corresponding 𝑦-output values. For example, in relation A, the input 𝑥 equals negative three gives an output 𝑦 equals six. In relation B, when 𝑥 is equal to negative two, 𝑦 is equal to six.

In order to answer this question, we need to recall our definition of a function. A function is a relation where each input value has exactly one output value. The keywords here are exactly one. In relation A, we have five distinct inputs: negative three, zero, three, eight, and negative 10. Each of these has exactly one output value: the numbers six, eight, 20, four, and eight, respectively. The fact that eight appears twice in the output 𝑦 does not matter. The key to any function is that each input, 𝑥, has exactly one value, 𝑦. In relation A, the input zero is only connected to eight and the input negative 10 is only connected to eight. We can, therefore, conclude that relation A is definitely a function.

Let’s now consider why relation B is not a function. When looking at our input values for relation B, we notice that the number negative two appears twice. The input negative two gives us an output of six, but it also gives us an output of 20. This means it does not have exactly one output value. Relation B is, therefore, not a function, confirming that the correct answer is relation A.

Our next question, we’ll look at coordinates or ordered pairs. As you can see, this question is very similar to the previous one. However, this time our data has been written as ordered pairs.

Which of the following relations represents a function? Is it relation A, four, 12; four, 15; five, 18; five, 21; and six, 24? Or is it relation B, four, 12; five, 15; six, 18; seven, 21; and eight, 24?

We recall that for a relation to be a function, each 𝑥-value or input must have exactly one corresponding 𝑦-value or output. Let’s firstly consider relation A. We should immediately notice here that the input or 𝑥-value four appears twice. Likewise, the 𝑥-value five appears twice. The 𝑥-value four is connected to the 𝑦-value 12 and the 𝑦-value 15. And the 𝑥-value five is connected to the 𝑦-value 18 and the 𝑦-value 21. This means that each 𝑥-value does not have exactly one 𝑦-value. This means that relation A is not a function.

In relation B, on the other hand, we have five unique 𝑥-values. The numbers four, five, six, seven, and eight. These are connected to one 𝑦-value, the numbers 12, 15, 18, 21, and 24, respectively. As each 𝑥-value has exactly one 𝑦-value, the correct answer is relation B. This represents a function.

In our penultimate question, we need to determine which set of ordered pairs are functions.

Which of the relations shown by the set of ordered pairs below does not represent a function? Is it (A) three, 11; 11, 19; 27, 35; and 43, 43. (B) Three, 11; four, 11; five, 11; and six, 11. (C) Negative eight, four; negative nine, four; 10, four; and 11, four. Or (D) three, four; 11, eight; three, 12; and 11, 11?

We recall that a function is a relation that connects every 𝑥-value with exactly one 𝑦-value. This means that within a relation of ordered pairs, each 𝑥-value must only appear once. In option (A), we have four distinct or unique 𝑥-values, the numbers three, 11, 27, and 43. This means that option (A) is a function. Option (B) also has four unique 𝑥-values, the numbers three, four, five, and six. The fact that each of the 𝑦-values is 11 does not impact on it being a function. Every 𝑥-value is connected with exactly one 𝑦-value. Therefore, (B) is also a function.

Option (C) is a function for exactly the same reason. We have four 𝑥-values, negative eight, negative nine, 10, and 11, all connecting to one 𝑦-value of four. This suggests that option (D) is not a function. We know this to be true because the input or 𝑥-value three connects to the 𝑦-value four and the 𝑦-value 12. Likewise, the 𝑥-value 11 connects to the 𝑦-values eight and 11. Option (D) is, therefore, not a function as each 𝑥-value does not connect to exactly one 𝑦-value. The correct answer is option (D).

Our final question involves graphed relations.

Which of the following relations represents a function given that 𝑥 is the input and 𝑦 is the output?

We recall that a function is a relation where each input has exactly one output. We are told that the input is 𝑥 and the output is 𝑦. Therefore, every 𝑥-value must have exactly one corresponding 𝑦-value. On our first graph, we can see that when 𝑥 is equal to one, we have two 𝑦-values, the values negative one and one. Likewise, when 𝑥 is equal to four, 𝑦 can be equal to two or negative two. The same is true when 𝑥 is equal to nine. The ordered pairs nine, negative three and nine, three both lie on the graph. This means that graph (A) is not a function as each input or 𝑥-value does not have exactly one output or 𝑦-value.

Option (B), on the other hand, is a function. It is actually a linear function. Each 𝑥- or input value has exactly one 𝑦- or output value. For example, when 𝑥 is equal to two, 𝑦 is equal to eight and when 𝑥 is equal to negative six, 𝑦 is equal to negative eight. Whichever value of 𝑥 we choose, there will be exactly one 𝑦-value. The correct answer is, therefore, graph (B).

We will now summarize the key points from this video. We found out in this video that a function is a special type of relation between two sets. A relation is a function if (i) for every element of the input set, there is an output and (ii) no element of the input set is mapped to more than one element of the output. This can be summarized by saying that a function is a relation where each object of the input connects to exactly one object of the output. We saw in this video that functions can be represented as diagrams, ordered pairs, in tables, or in graphs.