Question Video: Determining the Parity of a Piecewise-Defined Function Mathematics • Second Year of Secondary School

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.