Video Transcript
Determine the domain of the
function represented in the graph. What is the range of the
function?
In this question, we’re given the
graph of a function, and we need to determine its domain and range. Let’s start by recalling what we
mean by the domain of a function. The domain of any function is the
set of all input values for that function. And we need to determine this set
by using the graph of the function. And to do this, we need to recall
when we sketch the graph of a function, the 𝑥-coordinate of any point on our curve
tells us the input value for our function and the 𝑦-coordinate tells us the output
value. For example, we can see that our
curve passes through the point with coordinates three, two. The 𝑥-coordinate tells us the
input to the function, and the 𝑦-coordinate tells us the output. So 𝑓 evaluated at three is equal
to two. In particular, this tells us that
three is in the domain of our function.
Another way of thinking about this
is to consider the vertical line 𝑥 is equal to three. We can see there’s one point of
intersection between this vertical line and our function. Because there’s one point of
intersection between the vertical line and our curve, there’s one point on the curve
of our function with 𝑥-coordinate three. So it must be in the domain of our
function. We can do this with other values of
𝑥. For example, if we sketch the
vertical line 𝑥 is equal to negative one, we can see there’s a single point of
intersection between our curve and the vertical line. Therefore, negative one is also in
the domain of our function.
We can notice any vertical line
between negative one and three will intersect our curve once. And in fact, since our curve is
going to continue infinitely in either direction, any vertical line will intersect
the curve once. And therefore, all values of 𝑥
will be in the domain of our function. The domain of the function given in
the graph is the set of real numbers ℝ.
Let’s clear some space and move on
to the second part of our question. We need to determine the range of
this function. And we can recall the range of any
function is the set of all output values for the function given the domain or the
set of input values. And we can find the range of this
function from its graph in a very similar way.
Remember, the 𝑥-coordinate of any
point on our curve tells us a possible input value of our curve. And the corresponding 𝑦-coordinate
tells us the output for that value of 𝑥. For example, because our curve
passes through the point with coordinates two, three, we can conclude our function
evaluated at two must be equal to three. Three is in the range of our
function.
Therefore, another way of thinking
of the range of a function is the set of all 𝑦-coordinates of points which lie on
its curve. And we can determine this from its
graph. For example, we can see the line 𝑦
is equal to three intersects our curve once. And because there’s a point of
intersection, there is a point on the curve with 𝑦-coordinate three. So three is in the range of our
function. And we can try this with other
values of 𝑦 as well. For example, we can sketch the
horizontal line 𝑦 is equal to negative two. This time, we can see there’s two
points of intersection between our line and our curve. And it doesn’t matter that there’s
more than one point of intersection because all we’re interested in is negative two,
a possible output of our function. And the answer is yes, we found two
possible input values of 𝑥 which give us an output of negative two. So negative two is in the range of
our function.
However, this is not true for all
values of 𝑦. For example, if we sketch the line
𝑦 is equal to five, we can see there’s no point of intersection between our curve
and line. Graphically, we can see why this is
true. This is because three is the
largest output of our function. And because the graph of our
function continues towards negative ∞ in both directions, our function is
unbounded. So any value less than or equal to
three will be in the range of our function. And we can write this range in set
notation, where we need to use a square parenthesis on the right side of this
expression because three is in the range of our function.
This then gives us our final
answer. The domain of the function
represented in the graph is the set of all real numbers. And the range of the function is
the set of all real numbers less than or equal to three.
