Video Transcript
Discuss the monotonicity of the
following function on its domain.
In this question, we’re given the
graph of a function. And we need to use this to
determine the monotonicity of the function on its domain. To do this, let’s start by
recalling what we mean by the monotonicity of a function. It’s the intervals where the
function is increasing and the intervals where it’s decreasing. And we want to determine this on
the entire domain of the function. Remember, that’s the set of all
input values for the function. And we can see the domain of this
function from the graph. The 𝑥-coordinates of the points on
the curve will be the input values for our function. Since the graph of the function
ends with arrows in both sides, we know that this graph continues infinitely in both
directions. In particular, this tells us any
vertical line will intersect the graph once. So this function has a domain of
all real numbers.
Let’s now use the graph of this
function to determine the monotonicity of the function. Let’s start by determining the
intervals where our function is increasing. We recall the function will be
increasing when taking a larger input will result in a larger output. Another way of saying this is the
function slopes upwards. For example, by looking at the
graph, we can see the function is sloping upwards on this section. If we take any input value on the
section and then take another input value with a higher value of 𝑥, we can see the
output is larger. And it’s worth noting this still
holds true if we continue our graph. We can see the 𝑥-value where our
function stops increasing. This is when 𝑥 is negative
two. Therefore, the function is
increasing on the open interval from negative ∞ to negative two.
But this is not the only interval
where our function is sloping upwards. We can also see from the diagram
the same is true on the following interval. We can see that the 𝑥-values for
this section of the graph starts from one and ends at four. So we can also say that our
function is increasing on the open interval from one to four. There are no more sections where
our function is increasing. So if we take the union of these
two sets, we have all of the intervals where this function is increasing. We can follow the exact same
process to determine the intervals where this function is decreasing. And this will be the intervals
where the function slopes downwards. We can mark these on the
diagram.
The first section of the graph
which is decreasing starts at negative two and ends at one. So we can say our function is
decreasing on the open interval from negative two to one. Similarly, the second section of
our function that’s decreasing starts at four and continues indefinitely. We can say that our function is
decreasing on the open interval from four to ∞. Taking the union of these two sets
will give us all of the intervals on which the function is decreasing. This then gives us our final
answer. The function given in the diagram
is increasing on the open interval from negative ∞ to negative two union the open
interval from one to four and is decreasing on the open interval from negative two
to one union the open interval from four to ∞.
