Video Transcript
Determine the domain of the function 𝑓 of 𝑥 is equal to 𝑥 plus four when 𝑥 is in
the closed interval from negative four to four and 𝑓 of 𝑥 is equal to negative
eight 𝑥 plus 40 when 𝑥 is in the left-open and right-closed interval from four to
five.
In this question, we’re asked to determine the domain of a function 𝑓 of 𝑥. We’re given the piecewise function 𝑓 of 𝑥, and we’re also given a sketch of this
function. In fact, we can use either to determine the domain of the function 𝑓 of 𝑥. Let’s start by recalling what we mean by the domain of a function. We recall the domain of a function is the set of inputs of that function. Let’s start by finding the domain of our function from its sketch. Let’s start by discussing what we mean by a point on the curve.
If we take any point on our curve, then it has the form 𝑥, 𝑓 of 𝑥, where 𝑥 is the
input of our function and 𝑓 of 𝑥 is the output of our function. Therefore, the 𝑥-coordinate of all of the points on our curve represent the possible
inputs of our function. Therefore, we just need to find the 𝑥-coordinates of all of the points on our
curve. From the diagram, we can see that these range from negative four to five. And because we don’t have hollow dots at the end of our sketch, that means the
endpoints are included, so we need to include negative four and five. We can represent this in interval notation as the closed interval from negative four
to five. But this is not the only way we could have found the domain of our function. We could have also found the domain of our function from its piecewise
definition.
Recall a piecewise function has multiple subfunctions. For example, 𝑓 of 𝑥 has two subfunctions, the function 𝑥 plus four and the
function negative eight 𝑥 plus 40. And each subfunction is defined over its own subdomain. The subdomain for each function tells us the values of 𝑥 we need to input to take
that function as our output. For example, if we take an input value of 𝑥 in the closed interval from negative
four to four, then we use the function 𝑓 of 𝑥 as 𝑥 plus four. Therefore, since these subdomains are telling us the input values of our function 𝑓
of 𝑥, we can use these to determine the domain of 𝑓 of 𝑥.
The domain of 𝑓 of 𝑥 will be the union of the subdomains. Well, it’s worth pointing out, we might need to check that our function is in fact
defined over all of its subdomains. In this case, we don’t need to check because both subfunctions are linear, which
means they’ll be defined for all real values of 𝑥. Therefore, the domain of 𝑓 of 𝑥 will be the union of the subdomains. It’s the union of the closed interval from negative four to four and the left-open
right-closed interval from four to five. This means we take all 𝑥-inputs from negative four up to four inclusive. And we also include all of the values from four up to five inclusive. This is, of
course, just equal to the closed interval from negative four to five.
Therefore, we were able to show the domain of the function 𝑓 of 𝑥 is equal to 𝑥
plus four when 𝑥 is in the closed interval from negative four to four and 𝑓 of 𝑥
is equal to negative eight 𝑥 plus 40 when 𝑥 is in the left-open right-closed
interval from four to five is the closed interval from negative four to five.
