### 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 π sub three from the
first set exhibits this relationship to the object π sub seven in the second set,
then it can be represented with a relation diagram. We draw an arrow from π sub three
to π sub seven. Letβs now consider a real-life
example.

The following diagram represents
the relationship βson or daughter of.β 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 Jenniferβs father, and
Daniel is Noahβs 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 βlives inβ 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 π₯- or input values of
four, five, and eight and the π¦-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) π to one and π to
three; option (B) π to one, π to two, π to two, and π to three; or option (C) π
to three, π to one, and π 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 π 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 π and
π are connected to exactly one object in the second set; π connects to one and π
connects to two. However, the letter π 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 π is
only connected to the number three. The letter π is only connected to
the number one. Finally, the letter π is only
connected to the number three. Each object in the first set, the
letters π, π, and π, 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 π₯ and output π¦? Both relation A and relation B have
five π₯-input values and five corresponding π¦-output values. For example, in relation A, the
input π₯ equals negative three gives an output π¦ equals six. In relation B, when π₯ is equal to
negative two, π¦ 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 π¦ does not matter. The key to any function is that
each input, π₯, has exactly one value, π¦. 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.

Letβs 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, weβll 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 π₯-value or input must have exactly one corresponding π¦-value or
output. Letβs firstly consider relation A. We should immediately notice here that the input or π₯-value four appears twice. Likewise, the π₯-value five appears
twice. The π₯-value four is connected to
the π¦-value 12 and the π¦-value 15. And the π₯-value five is connected
to the π¦-value 18 and the π¦-value 21. This means that each π₯-value does
not have exactly one π¦-value. This means that relation A is not a
function.

In relation B, on the other hand,
we have five unique π₯-values. The numbers four, five, six, seven, and eight. These are connected to one
π¦-value, the numbers 12, 15, 18, 21, and 24, respectively. As each π₯-value has exactly one
π¦-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 π₯-value with exactly one π¦-value. This means that within a relation
of ordered pairs, each π₯-value must only appear once. In option (A), we have four
distinct or unique π₯-values, the numbers three, 11, 27, and 43. This means that option (A) is a
function. Option (B) also has four unique
π₯-values, the numbers three, four, five, and six. The fact that each of the π¦-values
is 11 does not impact on it being a function. Every π₯-value is connected with
exactly one π¦-value. Therefore, (B) is also a
function.

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

Our final question involves graphed
relations.

Which of the following relations
represents a function given that π₯ is the input and π¦ 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 π₯
and the output is π¦. Therefore, every π₯-value must have
exactly one corresponding π¦-value. On our first graph, we can see that
when π₯ is equal to one, we have two π¦-values, the values negative one and one. Likewise, when π₯ is equal to four,
π¦ can be equal to two or negative two. The same is true when π₯ 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 π₯-value does not have exactly one output or π¦-value.

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