### Video Transcript

Given that π is the function that maps positive real numbers onto the set of real numbers, where π of π₯ is π₯ minus 19. And π is the set that maps numbers in the closed interval from negative two to 13 onto the set of real numbers, where π of π₯ is π₯ minus six. Evaluate π times π of seven.

Letβs begin by recalling a little bit of notation. π dot π of π₯ or π times π of π₯ simply means we need to multiply the function π of π₯ by the function π of π₯; weβre finding their product. So, letβs just find π of π₯ times π of π₯. And weβll consider the domain in a moment.

Weβre told that π of π₯ is equal to π₯ minus 19 and π of π₯ is π₯ minus six. So, the product of these two functions is π₯ minus 19 times π₯ minus six. Usually, we might look to distribute these parentheses, but weβre not going to worry about that just yet since weβre going to be evaluating this function at a point. Before we do, though, we should check the domain of the function.

The domain of our function π is the set of positive real numbers. Whereas the domain, remember, thatβs the input of the function π, is the values of π₯ in the closed interval from negative two to 13. We know that the domain of our function must be the intersection of the domains of π and π. Thatβs the overlap. And so, itβs the set of positive real numbers up to and including 13. π₯ can be greater than zero and less than or equal to 13.

Now, weβre looking to evaluate π times π of seven, in other words, the value of our function when π₯ is equal to seven. π₯ is indeed in our domain, so weβre able to evaluate it. To do so, we replace π₯ with seven. And we find that π times π of seven is seven minus 19 times seven minus six. Seven minus 19 is negative 12, and seven minus six is one. So, π times π of seven is negative 12 times one, which is negative 12.

And so, weβve evaluated the product of our functions π and π at a given value of seven. Itβs negative 12.