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.
