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