### Video Transcript

Determine whether the function π of π₯ equals nine π₯ cubed is even, odd, or neither even or odd, given that π is defined for values of π₯ greater than negative seven and less than seven and π₯ is a real number.

Remember, a function is defined to be even if π of negative π₯ is equal to π of π₯ for all values of π₯ in the domain of π of π₯. And itβs said to be odd if π of negative π₯ is equal to negative π of π₯. Of course, a function is neither odd nor even if neither of these definitions hold. In our question, we see that π of π₯ is equal to nine π₯ cubed. So whatβs π of negative π₯? Well, if we replace π₯ with negative π₯, we see π of negative π₯ is nine times negative π₯ cubed. Well, negative π₯ cubed is negative π₯ cubed. So π of negative π₯ is negative nine π₯ cubed.

And so far, so good. π of negative π₯ does indeed look like itβs equal to negative π of π₯, making this an odd function. However, weβre going to need to be a little bit careful with the domain of our function. We see that π₯ takes values greater than negative seven and less than or equal to seven. So we evaluate π of seven. Itβs nine times seven cubed, which is 3087. But negative π of negative seven is not equal to π of seven. And this is because π of negative seven is not defined on the domain of π. π₯ equals negative seven is outside the domain of our function.

So weβve just shown that π of negative π₯ is not equal to negative π
of π₯ for all values of π₯ in the domain of our function and nor is π of negative π₯ equal to π of π₯. This means the function is neither odd nor even.