Question 1: Find are defined by:

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Question 2: Let Show that are both defined. Also, find

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Question 3: Let Show that is defined while is not defined. Also, find

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Question 4: Let and let and be two functions from to and from to respectively defined as: Show that and both are bijections and find and

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Question 5: Find when:

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Question 6: Let be the set of non-negative real numbers. If . Find Are they equal functions.

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Answer:

Therefore, the domains of are the same.

Answer:

Answer:

So, both have the same domains.

Now

Hence, the associative property has been verified.

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So, both have the same domains.

Now,

Question 11: Give examples of two functions such that is onto but is not onto.

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Let us consider given by

Therefore which is an identity function and hence it is onto.

Question 12: Give examples of two functions such that is injective but is not injective.

Answer:

We fist show that is not injective.

It can be observed that

Therefore is not injective.

Hence, is injective.

Question 13: If are one-one functions, show that is a one-one function.

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Question 14: If are onto functions show that is an onto function.

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