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 eight and 𝑓 of 𝑥 is equal to seven 𝑥 minus 63 when 𝑥 is in the
left-open, right-closed interval from eight to nine.
In this question, we’re asked to
find the domain of a given piecewise function 𝑓 of 𝑥. 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 𝑓 of 𝑥, each sub function has its own subdomain. And in this case, 𝑓 of 𝑥 has two
subfunctions: 𝑓 of 𝑥 is equal to 𝑥 plus four whenever 𝑥 is in the closed
interval from negative four to eight and 𝑓 of 𝑥 is equal to seven 𝑥 minus 63
whenever 𝑥 is in the left-open, right-closed interval from eight to nine. Therefore, all of the input values
of 𝑥 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 𝑓 of 𝑥 is the
union of its two subdomains.
And there’s a few different ways of
finding the union of these two sets. First, to say that 𝑥 is in the
closed interval from negative four to eight is the same as saying 𝑥 is greater than
or equal to negative four and 𝑥 is less than or equal to eight. We can say the same for our other
subdomain. To say that 𝑥 is in the left-open,
right-closed interval from eight to nine is the same as saying that 𝑥 is greater
than eight or less than or equal to nine. To say that 𝑥 is in the union of
these two sets is the same as saying either of these two conditions are true. And saying that 𝑥 is greater than
or equal to negative four and less than or equal to eight or 𝑥 is greater than
eight and less than or equal to nine is just the same as saying that 𝑥 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 𝑓 of 𝑥
is equal to 𝑥 plus four when 𝑥 is in the closed interval from negative four to
eight and 𝑓 of 𝑥 is equal to seven 𝑥 minus 63 when 𝑥 is in the left-open,
right-closed interval from eight to nine is the closed interval from negative four
to nine.
