# Question Video: Finding the Product of Two Polynomial Function and Stating Its Domain Mathematics • 9th Grade

Given that 𝑓₁: ℝ⁺ → ℝ such that 𝑓₁(𝑥) = 𝑥 − 4 and 𝑓₂: (−9,1] → ℝ such that 𝑓₂(𝑥) = 5𝑥 − 2, find (𝑓₁ ⋅ 𝑓₂)(𝑥) and state its domain.

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### Video Transcript

Given that 𝑓 sub one maps numbers from the set of positive real numbers onto the set of real numbers such that 𝑓 sub one of 𝑥 is equal to 𝑥 minus four and 𝑓 sub two maps numbers from the left-open, right-closed interval from negative nine to one onto the set of real numbers such that 𝑓 sub two of 𝑥 is equal to five 𝑥 minus two, find 𝑓 one times 𝑓 two of 𝑥 and state its domain.

Firstly, we recall that the product of 𝑓 one and 𝑓 two of 𝑥 is actually equal to the product of the two functions. It’s 𝑓 one of 𝑥 times 𝑓 two of 𝑥. And we know that when we combine functions, the domain of the resulting function will be the intersection of the domains of the respective functions. The only time we need to take into account additional information is if we’re finding the quotient and we need to make sure the denominator is not equal to zero. But of course, that’s not applicable here, so let’s begin by finding 𝑓 one times 𝑓 two of 𝑥. 𝑓 one is 𝑥 minus four and 𝑓 sub two is five 𝑥 minus two, so the product is 𝑥 minus four times five 𝑥 minus two.

Let’s distribute these parentheses by using the FOIL method. We’re going to multiply the first term in each binomial. 𝑥 times five 𝑥 is five 𝑥 squared. Next, we multiply the outer terms, which gives us negative two 𝑥. And we multiply the inner terms, giving us negative 20𝑥. Finally, we multiply the last term in each binomial. Negative four times negative two is eight. And so we see 𝑓 one times 𝑓 two of 𝑥 is five 𝑥 squared minus 22𝑥 plus eight. But what is the domain of this function?

Well, both 𝑓 one of 𝑥 and 𝑓 sub two of 𝑥 are, in fact, polynomials, and usually the domain of a polynomial is the set of real numbers. But we’re actually told that 𝑓 sub one maps numbers from the set of positive real numbers. And so that’s the domain of 𝑓 sub one. It’s the positive real numbers. Then we’re told that 𝑓 sub two maps numbers from the left-open, right-closed interval from negative nine to one. In other words, 𝑥 can be greater than negative nine and less than or equal to one. And so, that’s the domain of 𝑓 sub two. The domain of the product of our functions will be the intersection, the overlap, of these two domains. And so, let’s use a number line to establish where this overlap lies.

The domain of 𝑓 sub one is the set of positive real numbers. So that’s any number greater than zero as shown. The domain of 𝑓 sub two is values of 𝑥 greater than negative nine and less than or equal to one. The overlap lies here. It’s values of 𝑥 greater than zero and less than or equal to one. We can represent this domain using interval notation as shown. And so we see that 𝑓 one times 𝑓 two of 𝑥 is five 𝑥 squared minus 22𝑥 plus eight, and the domain is 𝑥 is an element of the left-open, right-closed interval from zero to one.