### Video Transcript

Determine the domain of the
function π of π₯ is equal to π₯ plus four when π₯ is in the closed interval from
negative four to eight and π of π₯ is equal to seven π₯ minus 63 when π₯ is in the
left-open, right-closed interval from eight to nine.

In this question, weβre asked to
find the domain of a given piecewise function π of π₯. And we can start by recalling the
domain of any function is the set of input values for that function. And for a piecewise-defined
function π of π₯, each sub function has its own subdomain. And in this case, π of π₯ has two
subfunctions: π of π₯ is equal to π₯ plus four whenever π₯ is in the closed
interval from negative four to eight and π of π₯ is equal to seven π₯ minus 63
whenever π₯ is in the left-open, right-closed interval from eight to nine. Therefore, all of the input values
of π₯ are defined by the subdomains. The union of the subdomains of a
piecewise-defined function is the domain of our function. So, the domain of π of π₯ is the
union of its two subdomains.

And thereβs a few different ways of
finding the union of these two sets. First, to say that π₯ is in the
closed interval from negative four to eight is the same as saying π₯ is greater than
or equal to negative four and π₯ is less than or equal to eight. We can say the same for our other
subdomain. To say that π₯ is in the left-open,
right-closed interval from eight to nine is the same as saying that π₯ is greater
than eight or less than or equal to nine. To say that π₯ is in the union of
these two sets is the same as saying either of these two conditions are true. And saying that π₯ is greater than
or equal to negative four and less than or equal to eight or π₯ is greater than
eight and less than or equal to nine is just the same as saying that π₯ is between
negative four and nine.

And since our inequalities at the
end of our expression are not strict, we need to use a closed interval. The domain of the function π of π₯
is equal to π₯ plus four when π₯ is in the closed interval from negative four to
eight and π of π₯ is equal to seven π₯ minus 63 when π₯ is in the left-open,
right-closed interval from eight to nine is the closed interval from negative four
to nine.