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math1a - quiz 8 - solns

# math1a - quiz 8 - solns - Math 1A Quiz 8 Solutions April...

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April 4th, 2008 1. a) Let f and g be two functions, and let a be a real number. What two conditions must be satisﬁed for a to be in the domain of f g ? b) What is the domain of f ( x ) = cosh - 1 ( 4 π sin - 1 ( x ) ) ? Solution to a): As discussed in section, the two criteria to check are: (a) a must be in the domain of g ( x ), and (b) g ( a ) must be in the domain of f ( x ). Solution to b): The domain of sin - 1 ( x ) is [ - 1 , 1], so the domain of 4 π sin - 1 ( x ) is also [ - 1 , 1]. The domain of cosh - 1 ( x ) is [1 , ). Ap- plying the criterion from part a), the domain of cosh - 1 ( 4 π sin - 1 ( x )) are those real numbers a such that: (a) a is in the domain of 4 π sin - 1 ( x ) (so - 1 a 1), and (b) 4 π sin - 1 ( a ) is in the domain of cosh - 1 ( x ) (so 4 π sin 1 ( a ) 1). The second condition says sin - 1 ( a ) π 4 ; equivalently, a sin( π 4 ) = 2 2 . So the answer is: the domain of cosh - 1 ( 4 π sin - 1 ( x )) are those

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math1a - quiz 8 - solns - Math 1A Quiz 8 Solutions April...

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