Video Transcript
What can be said about the
continuity of the function 𝑓 of 𝑥 is equal to nine 𝑥 squared plus four times the
cos squared of 𝑥. Option (A) 𝑓 of 𝑥 is continuous
on the set of real numbers because the map 𝑥 to the cos squared of 𝑥 is continuous
on the set of real numbers. Option (B) 𝑓 of 𝑥 is continuous
on the set of real numbers because the map from 𝑥 to nine 𝑥 squared is a
polynomial, and the map from 𝑥 to the cos squared of 𝑥 is continuous on the set of
real numbers. Or option (C) 𝑓 of 𝑥 is
continuous on the set of real numbers because the map from 𝑥 to nine 𝑥 squared is
a polynomial.
The question gives us a function 𝑓
of 𝑥, and it wants us to discuss the continuity of this function 𝑓 of 𝑥. We’re given three reasons why the
function 𝑓 of 𝑥 would be continuous on the set of real numbers. We need to decide which of these is
the correct reason 𝑓 of 𝑥 is continuous on the set of real numbers. Let’s start by taking a closer look
at our function 𝑓 of 𝑥. We can see it’s the sum of two
functions. It’s the sum of nine 𝑥 squared and
four times the cos squared of 𝑥. So if 𝑓 of 𝑥 is the sum of these
two functions, we could discuss the continuity of 𝑓 of 𝑥 by discussing the
continuities of its summands. In particular, we know the sum of
two continuous functions is itself continuous.
So let’s start by looking at nine
𝑥 squared. We know that nine 𝑥 squared is a
polynomial. And we know that all polynomials
are continuous on the set of real numbers. So the map from 𝑥 to nine 𝑥
squared is continuous on the set of real numbers because it’s a polynomial. Let’s now think about the second
term in our function 𝑓 of 𝑥, four times the cos squared of 𝑥. There’s a lot of different ways of
considering the continuity of this function. We can see in the answers that the
cosine is mentioned. Both of them consider the map from
𝑥 to the cos squared of 𝑥. This means the question wants us to
consider the continuity of the map from 𝑥 to the cos squared of 𝑥 and how this
relates to the continuity of four times the cos squared of 𝑥.
To do this, we’ll think of four
times the cos squared of 𝑥 as the product of two functions. It’s four times the cos squared of
𝑥. Next, we recall that the product of
two continuous functions is itself continuous, and we know that four is a continuous
function since it’s just a constant. So we now only need to discuss the
continuity of the map from 𝑥 to the cos squared of 𝑥. There’s a few different ways of
doing this. We recall all trigonometric
functions are continuous on their domain. And we know the cos of 𝑥 is
defined for all real values of 𝑥. This tells us that the map from 𝑥
to the cos of 𝑥 is continuous on the set of real numbers. And remember, we know the product
of two continuous functions is continuous.
So why don’t we take the product of
the cos of 𝑥 with the cos of 𝑥? Of course, this is the cos squared
of 𝑥. So this tells us that the map from
𝑥 to the cos squared of 𝑥 is continuous on the set of real numbers. So let’s recap all the information
we have here. We have the map from 𝑥 to the cos
squared of 𝑥 is continuous on the set of real numbers. And because of this, when we
multiply it by four, we still get a continuous function. Next, we have the map from 𝑥 to
nine 𝑥 squared is continuous on the set of real numbers because it’s a
polynomial. Finally, we know the sum of two
continuous functions is continuous. So because nine 𝑥 squared is
continuous and four cos squared of 𝑥 is continuous, their sum must be
continuous.
Therefore, we’ve shown the function
𝑓 of 𝑥 is continuous on the set of real numbers because the map from 𝑥 to nine 𝑥
squared is a polynomial and the map from 𝑥 to the cos squared of 𝑥 is continuous
on the set of real numbers.
