Video Transcript
Determine whether the function 𝑓
is even, odd, or neither, given that 𝑓 of 𝑥 is equal to negative nine 𝑥 minus
eight if 𝑥 is less than zero and nine 𝑥 minus eight if 𝑥 is greater than or equal
to zero.
Let’s recall the steps that allow
us to check the parity of a function, in other words, whether it’s even or odd. Firstly, we establish whether the
domain is centered at 𝑥 equals zero. If the answer to this is yes, then
we can say that it is even if 𝑓 of negative 𝑥 equals 𝑓 of 𝑥 and odd if 𝑓 of
negative 𝑥 equals negative 𝑓 of 𝑥. Well, we see by looking at the
values here that the domain of our function is simply the set of real numbers. And of course, that set is very
obviously centered at 𝑥 equals zero. And so we answer yes to the first
question. And so now we’re going to consider
the second criteria.
Since this is a piecewise function,
we need to be really careful. When 𝑥 is less than zero, we’re
working with 𝑓 of 𝑥 equals negative nine 𝑥 minus eight. And when it’s greater than or equal
to zero, we’re working with nine 𝑥 minus eight. And so let’s consider both parts of
our function individually. Let’s take negative nine 𝑥 minus
eight. And we’re now going to find 𝑓 of
negative 𝑥. It will be negative nine times
negative 𝑥 minus eight. But of course, a negative
multiplied by a negative is a positive. So 𝑓 of negative 𝑥 is equal to
nine 𝑥 minus eight.
Notice that that is equal to the
second part of our function. And it’s absolutely fine that it’s
equal to the other part, since we’ve changed the sign of the value of 𝑥. And so we’re moving on to the other
side of the piecewise function. And so for this part, yes, 𝑓 of
negative 𝑥 is equal to 𝑓 of 𝑥. So that part certainly appears to
be even. But let’s check the second
part. Let’s take 𝑓 of 𝑥 equals nine 𝑥
minus eight and then find 𝑓 of negative 𝑥. It’s nine times negative 𝑥 minus
eight, which is negative nine 𝑥 minus eight. We’ve changed the sign of the value
of 𝑥, and we’ve ended up with 𝑓 of 𝑥 when 𝑥 is less than zero. And so this part of the function is
also even. We can say that 𝑓 of negative 𝑥
is equal to 𝑓 of 𝑥. And we can therefore say that our
piecewise function is even.
