Video Transcript
In this video, we will learn how to
identify a function from a relation shown by a set of ordered pairs, a relation
diagram, an equation, or a graph.
We recall that a relation or
mapping transforms elements from one set onto elements from another. If every input in this mapping has
exactly one output, it is called a function. Let’s begin by defining this
formally. A function maps every element from
an input set onto exactly one element from an output set. Functions can either be one to one,
where one input has one output, or many to one, where many inputs map onto the same
output. If a function 𝑓 maps elements of
𝑥 to 𝑦, we can use the following notation. Functions can be represented by
ordered pairs of numbers, mapping diagrams, equations, and graphs. In our first example, we will
demonstrate how to apply this definition of a function to determine whether a set of
ordered pairs defines a function.
Which of the following relations
represents a function?
In this question, we are given two
relations 𝐴 and 𝐵, both of which consist of five ordered pairs. We begin by recalling that a
function is a rule that takes every member of a set and maps it onto exactly one
member in a second set. For a group of ordered pairs 𝑥,
𝑦, the input is 𝑥 and the output is 𝑦. This means that for a group of
ordered pairs to represent a function, no two ordered pairs can have the same input
with a different output. In other words, if an ordered pair
shares the same 𝑥-value, their 𝑦-value must also be the same.
We observe that relation 𝐴 has two
ordered pairs with an 𝑥-value of four. For relation 𝐴 to be a function,
the corresponding 𝑦-values in these ordered pairs must also be the same. However, the ordered pairs are
four, 12 and four, 15. Since the 𝑦-values are not the
same, relation 𝐴 cannot represent a function. Relation 𝐴 also has two ordered
pairs with an 𝑥-value of five. They too have different 𝑦-values:
the ordered pairs five, 18 and five, 21. We can therefore conclude that
relation 𝐴 does not represent a function.
The ordered pairs of relation 𝐵
have unique 𝑥-values: the integers four, five, six, seven, and eight. This satisfies the condition that a
function takes every member of a set and maps it onto exactly one member in the
second set. We can therefore conclude that the
relation that represents a function is relation 𝐵.
We could also represent these
ordered pairs on a mapping diagram. If we begin by considering relation
𝐴, we observe that elements four and five from the input or 𝑥-column map to more
than one element in the 𝑦- or output column. This mapping diagram therefore
cannot represent a function. For a mapping diagram to represent
a function, every input value must have a single output. When we consider relation 𝐵, we
see that every input, the integers four, five, six, seven, and eight, have a single
output, the integers 12, 15, 18, 21, and 24. Whilst it is not required for this
question, we notice that the 𝑦-values are three times the 𝑥-values. The function can therefore be
written as the equation 𝑦 is equal to three 𝑥.
In this first example, we have seen
that we can identify whether a relation is a function using ordered pairs or a
mapping diagram. Let’s now consider what happens
when a relation is represented by a graph. Since a function can be represented
by a set of ordered pairs, it follows that we can also use a graph as a visual
representation of a function. The graph of a function 𝑓 is
defined by a set of ordered pairs 𝑥, 𝑦 such that 𝑦 equals 𝑓 of 𝑥. For example, consider the function
𝑓 that maps 𝑥 onto 𝑥 squared. The set of ordered pairs that
represents this function is given by 𝑥, 𝑥 squared. And the graph of the function is
given by the graph of 𝑦 equals 𝑥 squared as shown.
This is an example of a many-to-one
function as there are values in the range of the function that are associated with
more than one value in the domain. For example, the points two, four
and negative two, four have the same 𝑦-value but a different 𝑥-value. To establish whether a graph
represents a function or not, we use the vertical line test. In order to carry out the vertical
line test, we begin by drawing a line parallel to the 𝑦-axis. If the vertical line intersects the
curve more than once, the graph does not represent a function. However, if the line does intersect
the curve exactly once for a given set of inputs, then the graph does represent a
function.
Let’s consider two examples. In the first figure, a vertical
line intersects the curve twice. This means that this graph does not
represent a function. However, in the second figure, any
vertical line will intersect the curve exactly once. This means that this graph does
represent a function. We will now look at an application
of the vertical line test.
Which of the following relations
represents a function given that 𝑥 is the input and 𝑦 is the output?
To answer this question, we will
use the vertical line test. This is a graphical way of working
out whether a graph represents a function. If a graph represents a function,
any vertical line will intersect the function at most once. We will therefore add a vertical
line to our diagrams and inspect the number of points of intersection with the
curves. In figure (A), we observe that if
we add the vertical line 𝑥 equals five, the line intersects the curve twice. This means that graph (A) cannot
represent a function. The vertical line 𝑥 equals
negative four intersects the curve of graph (B) exactly once. We can translate this vertical line
by any distance horizontally, and this will always be the case. We can therefore conclude that
graph (B) represents a function. The correct answer is (B).
Our definition of a function says
that it is some rule that maps an input onto exactly one output. In many cases, we can define that
rule algebraically. For example, graph (B) represents a
function of the graph 𝑦 equals two 𝑥 plus four. Let’s clear some space and see how
we can define this as a function. We can define the function using
functional notation as 𝑓 of 𝑥 is equal to two 𝑥 plus four. Alternatively, we can use arrow
notation. This means that we can determine
whether 𝑓 represents a function by plotting the graph of 𝑦 equals 𝑓 of 𝑥 and
performing the vertical line test.
Can the equation 𝑥 squared plus 𝑦
squared equals four be expressed as a function? If yes, state the function.
We recall that the vertical line
test can be used to determine whether a graph represents a function. This means that in order to
establish whether the equation 𝑥 squared plus 𝑦 squared equals four can be
expressed as a function, we can sketch its graph. Any equation in the form 𝑥 squared
plus 𝑦 squared equals 𝑟 squared represents a circle which is centered at the
origin with radius 𝑟. In this question, as 𝑟 squared is
equal to four and 𝑟 must be positive, the radius of our circle is two. Sketching the circle, we see that
it intersects the 𝑥- and 𝑦-axis at two and negative two.
Adding a vertical line to our
diagram, we note that it intersects the circle twice. As there is more than one point of
intersection, the graph does not represent a function. We can therefore conclude that the
equation 𝑥 squared plus 𝑦 squared equals four cannot be expressed as a
function. The correct answer is no.
In this question, it was fairly
easy to sketch the graph of the equation so we could perform the vertical line
test. However, we might have instead
attempted to rewrite the equation in terms of 𝑥. Subtracting 𝑥 squared from both
sides of our equation gives us 𝑦 squared is equal to four minus 𝑥 squared. Square rooting both sides of this
equation gives us 𝑦 is equal to positive or negative square root of four minus 𝑥
squared. We notice that when we take the
square root of both sides of the equation, we take the positive and negative square
root of the expression four minus 𝑥 squared. This means that for any single
value of 𝑥, there could be two possible outputs, a positive and a negative. Since a function maps every element
from one set onto exactly one element from a second set, we can deduce that the
equation 𝑥 squared plus 𝑦 squared equals four cannot represent a function.
In our final example, we will
demonstrate how to perform a similar process to represent an equation in function
notation.
Which of the following is the
equation 𝑦 cubed equals 𝑥 squared plus one expressed in function notation? Is it (A) 𝑓 of 𝑥 equals 𝑥
squared plus one? (B) 𝑓 of 𝑥 equals 𝑥 squared plus
one all cubed. Is it (C) this cannot be expressed
as a function? (D) 𝑓 of 𝑥 equals the cube root
of 𝑥 squared plus one. Or (E) 𝑓 of 𝑦 is equal to 𝑥
squared plus one.
To write any equation in function
notation, we need to see whether we can express 𝑦 as a function of 𝑥 such that 𝑦
is equal to 𝑓 of 𝑥. We will therefore rearrange the
equation to make 𝑦 the subject. In order to do this, we will cube
root both sides of the equation. This gives us 𝑦 is equal to the
cube root of 𝑥 squared plus one. We observe that 𝑦 is now written
as a function of 𝑥. In other words, to find the value
of 𝑦, we can substitute a value of 𝑥 into the expression the cube root of 𝑥
squared plus one. To express 𝑦 cubed equals 𝑥
squared plus one in function notation, we have 𝑓 of 𝑥 is equal to the cube root of
𝑥 squared plus one. The correct answer is option
(D).
It is important to note that
rearranging an equation to write 𝑦 as a function of 𝑥 will not always result in a
function. We must ensure that any 𝑥-value
from the domain of that function gives us a unique 𝑦-value when substituted in. For this reason, any rearrangement
that results in needing to invert an even exponent, for example, the square root or
fourth root, will not result in a function.
We will now finish this video by
recapping some key points. A function maps every element from
an input set onto exactly one element from an output set. Functions can either be one to one,
one input has one output, or many to one, where many inputs map onto the same
output. We have seen that functions can be
represented by ordered pairs of numbers, mapping diagrams, equations, and
graphs. The function notation 𝑦 equals 𝑓
of 𝑥 means that 𝑦 is a function of 𝑥. The arrow notation shown means that
the function 𝑓 maps elements of set 𝑥 onto an element of set 𝑦. We saw in this video that there
were several ways to test whether a relation is a function. One way to do this graphically was
using the vertical line test.
