Video: Continuity of Trigonometric Functions

Find the set on which 𝑓(𝑥) = (5𝑥 − 5) cos 5𝑥 is continuous. [A] 𝑓(𝑥) is continuous on ℝ because 𝑥 ↦ (5𝑥 − 5) is a polynomial, and 𝑥 ↦ (cos 5𝑥) is continuous on ℝ. [B] 𝑓(𝑥) is continuous on ℝ − {5}. [C] 𝑓(𝑥) is continuous on ℝ because 𝑥 ↦ (cos 5𝑥) is continuous on ℝ. [D] 𝑓(𝑥) is continuous on ℝ because 𝑥 ↦ (5𝑥 − 5) is a polynomial.

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Video Transcript

Find the set on which the function 𝑓 of 𝑥 is equal to five 𝑥 minus five multiplied by the cos of five 𝑥 is continuous. A) 𝑓 of 𝑥 is continuous on the set of real numbers because the map from 𝑥 to five 𝑥 minus five is a polynomial and the map from 𝑥 to the cos of five 𝑥 is continuous on the set of real numbers. Option B) 𝑓 of 𝑥 is continuous on the set of real numbers excluding five. Option C) 𝑓 of 𝑥 is continuous on the set of real numbers because the map of 𝑥 to the cos of five 𝑥 is continuous on the set of real numbers. And option D) 𝑓 of 𝑥 is continuous on the set of real numbers because the map from 𝑥 to five 𝑥 minus five is a polynomial.

We can start by recalling if two functions 𝑔 of 𝑥 and ℎ of 𝑥 are continuous on their domains, then the function 𝑔 of 𝑥 multiplied by ℎ of 𝑥 is continuous on the domain of 𝑔 multiplied by ℎ. We can see that the function 𝑓 of 𝑥 given to us in the question is equal to the product of two functions. It’s the product of five 𝑥 minus five and the cos of five 𝑥. In particular, we can see that our function 𝑓 of 𝑥 is the product of a polynomial and a trigonometric function.

Using this rule, we can use the continuity of our functions 𝑔 of 𝑥 and ℎ of 𝑥 individually to deduce the continuity of their product. First, we can recall that every polynomial is continuous on the entire set of real numbers. In particular, five 𝑥 minus five is a polynomial. So, we can conclude that five 𝑥 minus five is continuous on the set of real numbers because it is a polynomial.

Next, we recall that every trigonometric function is continuous on its domain. In particular, this means that we know that the cos of five 𝑥 is continuous on its domain. So, we need to find the domain of the cos of five 𝑥. Well, we know that for any real number 𝑥, the cos of 𝑥 is defined. And we can use this to argue that the cos of five 𝑥 is defined for any real number 𝑥.

Therefore, the set of reals is the domain of the function cos of five 𝑥. And we can conclude that the cos of five 𝑥 is continuous on the set of real numbers. Therefore, what we have shown is that the function 𝑓 of 𝑥 is the product of two functions which are both continuous on the real numbers.

Therefore, we can use this rule for continuity to conclude that 𝑓 of 𝑥 must be continuous on the set of real numbers. Therefore, what we have shown is that the function 𝑓 of 𝑥 is continuous on the real numbers because the map from 𝑥 to five 𝑥 minus five is a polynomial and the map from 𝑥 to cos of five 𝑥 is continuous on the real numbers.

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