# Video: Continuity of the Sum of a Polynomial and Trigonometric Function

What can be said about the continuity of the function 𝑓(𝑥) = 9𝑥² + 4 cos² 𝑥? A) 𝑓(𝑥) is continuous on ℝ because 𝑥 ↦ cos² 𝑥 is continuous on ℝ. B) 𝑓(𝑥) is continuous on ℝ because 𝑥 ↦ 9𝑥² is a polynomial, and 𝑥 ↦ cos² 𝑥 is continuous on ℝ. C) 𝑓(𝑥) is continuous on ℝ because 𝑥 ↦ 9𝑥² is a polynomial.

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### 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.