### Video Transcript

What can be said about the
continuity of the function π of π₯ is equal to nine π₯ squared plus four times the
cos squared of π₯. Option (A) π of π₯ is continuous
on the set of real numbers because the map π₯ to the cos squared of π₯ is continuous
on the set of real numbers. Option (B) π of π₯ is continuous
on the set of real numbers because the map from π₯ to nine π₯ squared is a
polynomial, and the map from π₯ to the cos squared of π₯ is continuous on the set of
real numbers. Or option (C) π of π₯ is
continuous on the set of real numbers because the map from π₯ to nine π₯ squared is
a polynomial.

The question gives us a function π
of π₯, and it wants us to discuss the continuity of this function π of π₯. Weβre given three reasons why the
function π of π₯ would be continuous on the set of real numbers. We need to decide which of these is
the correct reason π of π₯ is continuous on the set of real numbers. Letβs start by taking a closer look
at our function π of π₯. We can see itβs the sum of two
functions. Itβs the sum of nine π₯ squared and
four times the cos squared of π₯. So if π of π₯ is the sum of these
two functions, we could discuss the continuity of π of π₯ by discussing the
continuities of its summands. In particular, we know the sum of
two continuous functions is itself continuous.

So letβs start by looking at nine
π₯ squared. We know that nine π₯ squared is a
polynomial. And we know that all polynomials
are continuous on the set of real numbers. So the map from π₯ to nine π₯
squared is continuous on the set of real numbers because itβs a polynomial. Letβs now think about the second
term in our function π of π₯, four times the cos squared of π₯. Thereβs 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
π₯ to the cos squared of π₯. This means the question wants us to
consider the continuity of the map from π₯ to the cos squared of π₯ and how this
relates to the continuity of four times the cos squared of π₯.

To do this, weβll think of four
times the cos squared of π₯ as the product of two functions. Itβs four times the cos squared of
π₯. Next, we recall that the product of
two continuous functions is itself continuous, and we know that four is a continuous
function since itβs just a constant. So we now only need to discuss the
continuity of the map from π₯ to the cos squared of π₯. Thereβs a few different ways of
doing this. We recall all trigonometric
functions are continuous on their domain. And we know the cos of π₯ is
defined for all real values of π₯. This tells us that the map from π₯
to the cos of π₯ is continuous on the set of real numbers. And remember, we know the product
of two continuous functions is continuous.

So why donβt we take the product of
the cos of π₯ with the cos of π₯? Of course, this is the cos squared
of π₯. So this tells us that the map from
π₯ to the cos squared of π₯ is continuous on the set of real numbers. So letβs recap all the information
we have here. We have the map from π₯ to the cos
squared of π₯ 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 π₯ to
nine π₯ squared is continuous on the set of real numbers because itβs a
polynomial. Finally, we know the sum of two
continuous functions is continuous. So because nine π₯ squared is
continuous and four cos squared of π₯ is continuous, their sum must be
continuous.

Therefore, weβve shown the function
π of π₯ is continuous on the set of real numbers because the map from π₯ to nine π₯
squared is a polynomial and the map from π₯ to the cos squared of π₯ is continuous
on the set of real numbers.