Question Video: Evaluating the Product of Two Functions at a Given Value Mathematics

Given that 𝑓: ℝ⁺ β†’ ℝ, where 𝑓(π‘₯) = π‘₯ βˆ’ 19, and 𝑔: [βˆ’2, 13] β†’ ℝ, where 𝑔(π‘₯) = π‘₯ βˆ’ 6, evaluate (𝑓 β‹… 𝑔)(7).


Video Transcript

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

Let’s begin by recalling a little bit of notation. 𝑓 dot 𝑔 of π‘₯ or 𝑓 times 𝑔 of π‘₯ simply means we need to multiply the function 𝑓 of π‘₯ by the function 𝑔 of π‘₯; we’re finding their product. So, let’s just find 𝑓 of π‘₯ times 𝑔 of π‘₯. And we’ll consider the domain in a moment.

We’re told that 𝑓 of π‘₯ is equal to π‘₯ minus 19 and 𝑔 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 we’re not going to worry about that just yet since we’re 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 𝑓 is the set of positive real numbers. Whereas the domain, remember, that’s the input of the function 𝑔, 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 𝑓 and 𝑔. That’s the overlap. And so, it’s 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, we’re looking to evaluate 𝑓 times 𝑔 of seven, in other words, the value of our function when π‘₯ is equal to seven. π‘₯ is indeed in our domain, so we’re able to evaluate it. To do so, we replace π‘₯ with seven. And we find that 𝑓 times 𝑔 of seven is seven minus 19 times seven minus six. Seven minus 19 is negative 12, and seven minus six is one. So, 𝑓 times 𝑔 of seven is negative 12 times one, which is negative 12.

And so, we’ve evaluated the product of our functions 𝑓 and 𝑔 at a given value of seven. It’s negative 12.

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