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

What can be said about the continuity of the function 𝑓(π‘₯) = 9π‘₯Β² + 4 cosΒ² π‘₯? A) 𝑓(π‘₯) is continuous on ℝ because π‘₯ ↦ cosΒ² π‘₯ is continuous on ℝ. B) 𝑓(π‘₯) is continuous on ℝ because π‘₯ ↦ 9π‘₯Β² is a polynomial, and π‘₯ ↦ cosΒ² π‘₯ is continuous on ℝ. C) 𝑓(π‘₯) is continuous on ℝ because π‘₯ ↦ 9π‘₯Β² is a polynomial.

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Video Transcript

What can be said about the continuity of the function 𝑓 of π‘₯ is equal to nine π‘₯ squared plus four times the cos squared of π‘₯. Option (A) 𝑓 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) 𝑓 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) 𝑓 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 𝑓 of π‘₯, and it wants us to discuss the continuity of this function 𝑓 of π‘₯. We’re given three reasons why the function 𝑓 of π‘₯ would be continuous on the set of real numbers. We need to decide which of these is the correct reason 𝑓 of π‘₯ is continuous on the set of real numbers. Let’s start by taking a closer look at our function 𝑓 of π‘₯. We can see it’s the sum of two functions. It’s the sum of nine π‘₯ squared and four times the cos squared of π‘₯. So if 𝑓 of π‘₯ is the sum of these two functions, we could discuss the continuity of 𝑓 of π‘₯ 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 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 it’s a polynomial. Let’s now think about the second term in our function 𝑓 of π‘₯, four times the cos squared of π‘₯. 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 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, we’ll think of four times the cos squared of π‘₯ as the product of two functions. It’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 it’s just a constant. So we now only need to discuss the continuity of the map from π‘₯ to the cos squared of π‘₯. 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 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 don’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 let’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 it’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, we’ve shown the function 𝑓 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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