# Question Video: Determining the Parity of a Piecewise-Defined Function Mathematics

Determine whether the function π is even, odd, or neither, given that π(π₯) = β9π₯ β 8 if π₯ < 0, and π(π₯) = 9π₯ β 8 if π₯ β©Ύ 0.

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### 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.

Letβs recall the steps that allow us to check the parity of a function, in other words, whether itβs 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 weβre going to consider the second criteria.

Since this is a piecewise function, we need to be really careful. When π₯ is less than zero, weβre working with π of π₯ equals negative nine π₯ minus eight. And when itβs greater than or equal to zero, weβre working with nine π₯ minus eight. And so letβs consider both parts of our function individually. Letβs take negative nine π₯ minus eight. And weβre 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 itβs absolutely fine that itβs equal to the other part, since weβve changed the sign of the value of π₯. And so weβre 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 letβs check the second part. Letβs take π of π₯ equals nine π₯ minus eight and then find π of negative π₯. Itβs nine times negative π₯ minus eight, which is negative nine π₯ minus eight. Weβve changed the sign of the value of π₯, and weβve 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.