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Problem 32 Easy Difficulty

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ N(r) = \tan^{-1}(1 + e^{-r^2}) $

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Video Transcript

this is problem number thirty two of the sewer calculus eighth edition section two point five. Explain using terms four, five, seven and nine where the function is continuous. That every number in its domain state to state that I mean this function and our is equal to inverse tangent or arc tangent of the quantity one plus eed to the negative are squared. So we refer to our therms. We have a function within a function. This has to do with your nine. So as long as the function on the inside is continuous for its demon and the function, yes, it is also continues on its domain. Then the composite function of the two is also continuous. Within the Arc Tangent function we have on exponential function, this function is definitely continuous. That's so mean and in fact that's something his all real numbers since our is allowed to be any number. So there are no domain restrictions and then they are tendon function. This inverse tangent function is a form of it is a example of a triple trigon metric function that is continuous on all reals. That's the domain of this specific. You're gonna mention function here in seven states that any treatment function is continuous on its doing, but specifically our attention. Its domain specifically is negative. Infinity to infinity. So all rials on and as we discussed, both of these have no Domaine restrictions. So since this is continuous and all rials and our attention function also continues on Honoria, all rials, no domain restrictions means that the combined domain for this composite function it's going to be all real numbers from negative infinity. That's your final answer.