In this video, we will learn how to
identify the domain and range of functions from their graphs. First of all, we’ll think about the
definition of domain and range. If we let this machine represent
some function machine, the domain will be the values that we begin with. The domain is the complete set of
possible values, and these values are independent. It is the independent variable
value. And on a standard coordinate grid,
these are going to be the 𝑥-values. The 𝑥-axis represents the
independent variables. And the range is the complete set
of all possible resulting values. It is the dependent variable. And on a standard coordinate grid,
this is the 𝑦-value. The 𝑦-values are the output values
of this function. The 𝑥-values are the inputs, and
the 𝑦-values will be the output.
To explore this further, we’ll
start looking at some graphs and some example problems.
The domain of the function 𝑓 of 𝑥
The function 𝑓 of 𝑥 here is
represented by these five points. First, we remember that the domain
is the set of all possible 𝑥-values for a function. And then we recognize that on a
coordinate grid, the 𝑥-axis is the horizontal axis, which means the 𝑥-values of
this function will be found by looking at where these points fall horizontally. All the way to the left, we have a
point at negative seven. To the right, we have a point at
negative six, followed by negative five, negative four, and negative three.
It’s important to notice that these
points are not connected with the line. Because of that, we know that this
is not a continuous function and that the domain is then just going to be a list of
the possible 𝑥-values. In set notation, it would look like
this: negative seven, negative six, negative five, negative four, and negative
If we wanted to, we could consider
the range as well. The range will be the possible
𝑦-values of this function. And that will be how far the points
are located up or down, where they fall on the vertical axis. For this function, we have
𝑦-values of one, two, three, four, and five. And set notation for the range
would look like this: one, two, three, four, five.
As this question was only asking
for the domain, the domain of 𝑓 of 𝑥 here is the set negative seven, negative six,
negative five, negative four, and negative three.
Let’s look at another example.
Determine the domain and range of
the function 𝑓 of 𝑥 equals negative four.
In the image, we’ve been given the
graph of the function 𝑓 of 𝑥 equals negative four. In order to calculate the domain
and range, we remember that the domain is represented by the 𝑥-values and the range
is represented by the 𝑦-values on the graph. We also remember that the domain is
the independent variable. It’s the variable we plug in to our
function. We want to know what is the set of
values that 𝑥 can be.
Now, on this graph, it might look
like 𝑥 only goes from negative four to positive four. However, we recognize that this is
a function that continues in both directions. To the right, 𝑥 would continue out
to positive ∞ and to the left negative ∞. So how should we write this as a
We could use this symbol that looks
a little bit like an R. This symbol represents all real numbers. The domain for 𝑥 can be any real
What about the range? The range is a little bit different
here. The range will be the 𝑦-values,
that is, the distance up or down from zero. For every 𝑥-value in this
function, 𝑦 is always negative four. 𝑦 does not change. And that means the only outcome,
the only output of this function, is negative four. The range is the set of negative
four. And so we can say for the function
𝑓 of 𝑥 equals negative four, the domain is all real numbers and the range is the
set negative four.
In our next example, we’re given
the graph of a cubic function and we’ll need to find its domain and range.
Find the domain and range of the
function 𝑓 of 𝑥 equals 𝑥 minus one cubed in all reals.
We’ve already been given the graph
of this function, 𝑥 minus one cubed. So now we just need to think about
what the domain and range are. When we have a graph, the domain is
represented by the set of possible 𝑥-values and the range is the set of all
possible 𝑦-values. It’s important to know that when we
have this type of graph, we know that they continue in both directions. While we’re only seeing a bit of
this function, from 𝑥 negative two to 𝑥 positive three, we know that it continues
in both directions. The same thing is true for the
𝑦-values. We’re only seeing 𝑦-values up to
positive 10 and down to negative 10.
However, this function continues
outside of this window on our graph. In this case, we have no limits on
our domain or range. The domain can be all real numbers,
and the range can be all real numbers. It’s also possible that we might
want to write this in interval notation instead of in set notation. The interval of the domain would be
written as negative ∞ to ∞. And in this case, the same thing
will be true for the range interval, all real numbers or values from negative ∞ to
With the interval notation here,
it’s important to note that we use the round brackets when we are not including what
is on the end. So what these say is that we want
to go up to ∞ but not including ∞.
For our next example, we’ll look at
determining the domain and the range of a piecewise function.
Determine the domain of the
We know that the domain of this
function will be the set of all possible 𝑥-values. And on a coordinate grid, that is
the 𝑥-axis, the horizontal axis. We see designated values from
negative seven all the way to positive seven. However, we should know that the
arrows on either side of this graph indicate that this function continues. On the left, we would say that the
graph could continue to negative ∞ and on the right to positive ∞.
However, let’s think carefully
about what’s happening at zero. When 𝑥 equals zero, does this
function have a result? We know that it does because the
point is colored in at zero, four. Zero, four is a result, but zero,
negative four is not filled in and is therefore not a result of this function. Since we do have a result at zero,
we can confirm that there’s a domain of all real numbers.
This question hasn’t asked us for a
range. But if we wanted to add the range,
that would be the output values, the set of possible 𝑦-values. And we see that there are two
possible values: one value at four and one value at negative four. In set notation, we could write
that the range is therefore negative four and four. As the question has only asked us
to identify the domain, we can simply say that the domain is all reals.
In our final example, we’ll look at
a graph where there are limits to the domain and the range.
Find the domain and range of the
function 𝑓 of 𝑥 is equal to negative one over 𝑥 minus five.
We’ve already been given a graph of
this function. And we can use the graph to
identify both the domain and the range of the function. The domain will be the set of all
possible 𝑥-values. And on this graph, we can use the
𝑥-axis to identify that. And the range will be the set of
all possible 𝑦-values. We’ll use the 𝑦-axis to identify
But before we do that, let’s
carefully consider the behavior of the function in the graph we’re looking at. We see that it kind of has two
pieces: one above the 𝑥-axis and one below the 𝑥-axis. And then we have this dotted
line. When we have a dotted line like
this on the graph, it represents an asymptote of the function. An asymptote is a line that a curve
approaches as it heads towards ∞. The curve will never cross the
asymptote. And this asymptote is located at 𝑥
equal to five. And that means we can say, for
sure, that the domain does not include the value 𝑥 equals five.
But if we look at the rest of the
function, we can see that there are 𝑥-values extending in the left and right
direction. And so 𝑥 can be anything except
for positive five, which means the domain is all reals minus the set five. Now, if we’re thinking about the
range, we’re thinking about the vertical behavior of our graph. And again, we notice that there is
one piece of this graph above the 𝑥-axis and one piece below it. Even though they didn’t add a
dotted line, the 𝑥-axis represents another asymptote of this function. The 𝑦-value of this function is
getting closer and closer to zero, but it is never crossing zero. And that is true on both the left-
and the right-hand side of this function. This means that the 𝑦-values can
be anything except for zero.
And so, in a similar format, we say
that the range will be all reals minus the set zero. The set of five in the domain and
the set of zero in the range represents the vertical and horizontal asymptotes of
this function, and we’d correctly label the domain and range.
Before we finish, let’s review some
key points from this video. The domain of a function is the
complete set of possible values of the independent variable. The range of a function is the
complete set of possible resulting values. Given the graph of a function, the
domain is all possible 𝑥-values and the range is all possible 𝑦-values.