### Video Transcript

Determine whether the function π
is even, odd, or neither, given that π of π₯ is equal to negative nine π₯ minus
eight if π₯ is less than zero and nine π₯ minus eight if π₯ is greater than or equal
to zero.

Letβs recall the steps that allow
us to check the parity of a function, in other words, whether itβs even or odd. Firstly, we establish whether the
domain is centered at π₯ equals zero. If the answer to this is yes, then
we can say that it is even if π of negative π₯ equals π of π₯ and odd if π of
negative π₯ equals negative π of π₯. 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 π₯ equals zero. And so we answer yes to the first
question. And so now weβre going to consider
the second criteria.

Since this is a piecewise function,
we need to be really careful. When π₯ is less than zero, weβre
working with π of π₯ equals negative nine π₯ minus eight. And when itβs greater than or equal
to zero, weβre working with nine π₯ minus eight. And so letβs consider both parts of
our function individually. Letβs take negative nine π₯ minus
eight. And weβre now going to find π of
negative π₯. It will be negative nine times
negative π₯ minus eight. But of course, a negative
multiplied by a negative is a positive. So π of negative π₯ is equal to
nine π₯ minus eight.

Notice that that is equal to the
second part of our function. And itβs absolutely fine that itβs
equal to the other part, since weβve changed the sign of the value of π₯. And so weβre moving on to the other
side of the piecewise function. And so for this part, yes, π of
negative π₯ is equal to π of π₯. So that part certainly appears to
be even. But letβs check the second
part. Letβs take π of π₯ equals nine π₯
minus eight and then find π of negative π₯. Itβs nine times negative π₯ minus
eight, which is negative nine π₯ minus eight. Weβve changed the sign of the value
of π₯, and weβve ended up with π of π₯ when π₯ is less than zero. And so this part of the function is
also even. We can say that π of negative π₯
is equal to π of π₯. And we can therefore say that our
piecewise function is even.