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

What can be said about the continuity of the function π(π₯) = 9π₯Β² + 4 cosΒ² π₯? A) π(π₯) is continuous on β because π₯ β¦ cosΒ² π₯ is continuous on β. B) π(π₯) is continuous on β because π₯ β¦ 9π₯Β² is a polynomial, and π₯ β¦ cosΒ² π₯ is continuous on β. C) π(π₯) is continuous on β because π₯ β¦ 9π₯Β² is a polynomial.

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### 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₯. Weβre 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. Letβs start by taking a closer look at our function π of π₯. We can see itβs the sum of two functions. Itβs 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 letβs 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 itβs a polynomial. Letβs now think about the second term in our function π of π₯, four times the cos squared of π₯. Thereβ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 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, weβll think of four times the cos squared of π₯ as the product of two functions. Itβs 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 itβs just a constant. So we now only need to discuss the continuity of the map from π₯ to the cos squared of π₯. Thereβs 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 donβt 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 letβs 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 itβs 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, weβve 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.

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