State the domain and range of the
following function. 𝑓 of 𝑥 equals negative two for
negative 12 is less than 𝑥 which is less than or equal to negative one. 11 over two for negative one is
less than 𝑥 which is less than one. And zero for one is less than or
equal to 𝑥 which is less than or equal to four.
If we think of a function machine,
we remember that the domain will contain all of the values that can go into the
function and the range are all the values that can come out of the function. We have one function. And we’re given three
conditions. The first line tells us what to do
when 𝑥 is greater than negative 12 but less than or equal to negative one. This means that 𝑥 between negative
12 and equal to or less than negative one must be part of the domain. When 𝑥 falls in that domain, the
output is negative two. Which means negative two must be
part of our range.
Now, let’s consider the second set
of information. We have the case when 𝑥 falls
between negative one and one. This means we have an additional
domain of 𝑥 between negative one and one. That also means that the range
includes eleven-halves. This, of course, means that the
domain includes one is greater than or equal to 𝑥, which is greater than or equal
to four, and the range includes zero. We need to think about how we write
this. We could write the range in set
notation like this. The range is negative two,
eleven-halves, and zero.
If we look closely at our domain,
we’ll see that we have 𝑥-values beginning when 𝑥 is greater than negative 12 and
ending when 𝑥 is less than or equal to four. This is true because the domain can
include anything that can go into our function. And every value from something
greater than negative 12 all the way until positive four has an output value for
this function. And so, we say, for the given
function 𝑓 of 𝑥, its domain is such that negative 12 is less than 𝑥 which is less
than or equal to four. And the range includes negative
two, eleven-halves, and zero.