Video Transcript
Which of the following represents
the domain and range of the function in the given graph? Option (A) the domain is the
left-open, right-closed interval from negative ∞ to one, and the range is the set of
real numbers. Option (B) the domain is the set of
real numbers, and the range is the set of real numbers. Option (C) the domain is the set of
real numbers, and the range is the open interval from negative ∞ to one. Option (D) the domain is the set of
real numbers, and the range is the left-open, right-closed interval from negative ∞
to one. Or is it option (E) the domain is
the set of real numbers, and the range is the left-closed, right-open interval from
one to ∞?
In this question, we are asked to
find the domain and range of a function using its given graph. To do this, let’s start by
recalling what we mean by the domain and range of a function. We can recall that the domain of a
function is the set of all input values of the function and that the range of a
function is the set of all output values of the function, given its domain. We can find the domain and range of
a function from its graph by recalling how we graph a function. Every coordinate of a point on the
graph has 𝑥-coordinate equal to the input of the function, and the 𝑦-coordinate of
each point is the corresponding output of the function.
This means that the domain of a
function is the set of all 𝑥-coordinates of points which lie on its graph and the
range of a function is the set of all 𝑦-coordinates of points which lie on its
graph. Let’s start by finding the domain
of the function from its graph. We want to determine all of the
possible inputs of the function, and we can do this by considering vertical
lines. For example, we see that the
vertical line 𝑥 equals four intersects the graph of our function. This means that four is an input of
the function. In particular, the output of the
function at 𝑥 equals four is the 𝑦-coordinate of this point, negative seven.
If we continue this process for
more input values, we can notice that it appears that every possible value of 𝑥 is
an input of the function. For example, the vertical line 𝑥
equals negative six intersects the graph. To show that this is true, we
recall the graph will extend indefinitely in each direction. We can represent this with arrows
on the graph. Now, since any vertical line
intersects the graph, any real value of 𝑥 is a possible input value of the
function. Hence, the domain of the function
in the graph is the set of all real numbers. We can use this information to
eliminate option (A) since it does not include all real numbers in the domain.
Let’s now consider the range of the
function, that is, the set of all outputs of the function. We can find possible outputs of the
function by considering the 𝑦-coordinates of points on the graph. For example, we know that negative
seven is a possible output of the function since the horizontal line 𝑦 equals
negative seven intersects the graph. In the same way, the range of a
function is the set of all 𝑦-coordinates of points which lie on its graph. Therefore, we can determine the
range of a function from its graph by considering which horizontal lines intersect
the graph.
Let’s consider the horizontal lines
that intersect the graph. We can see that the highest
horizontal line that intersects the graph occurs at 𝑦 equals one. There are two things we need to
note about this. First, if we call our function 𝑓,
then we have shown that 𝑓 of zero equals one, so one is in the range of 𝑓. Second, no horizontal line above
one intersects the graph, so no value above one is in the range of the function. We can also note that any
horizontal line below 𝑦 equals one will intersect the graph. So all values less than or equal to
one are in the range of this function.
We want to write this as a set. And to do this, we will use
interval notation. We want to include all values less
than or equal to one. We do this by using a parenthesis
at negative ∞ to show that negative ∞ is not included and a square bracket at one to
show that one is an element of the range. We can see that this matches option
(D). Hence, the domain of the function
given in the graph is the set of all real numbers, and the range of the function
given in the graph is the left-open, right-closed interval from negative ∞ to
one.
