4. Your classmate is teaching you about constant functions, and gives the follow

Question

4. Your classmate is teaching you about constant functions, and gives the following definition: "A function h A- B is called a constant function if for all a E A we have h(a)-b for some element b B." (a) Explain carefully why this is not a good definition of a constant function. Give a correct definition. Let f : S ? T and g : T ? U be functions. Suppose that g of is a constant function. (b) What can you say about f given that g is one-to-one? Prove your claim. (c) What can you say about g given that f is onto? Prove your claim. (d) Suppose that U- S, so that go f is defined. Is it necessarily true that go f is a constant function?

Explanation / Answer

A function is said to be a constant function if its Range set is a Singleton set.

Or you can say for every input value we get the same output value.

For part (b) f need not be one to one and onto

take f(x)=1 for all x in S and g(x)= x for all x inT

Then gof is constant function.

For part ( c) interchange the roles of g and f in part (b)

g need not be one to one and onto.

For part (d) if we are not assuming that gof is a constant function.what we have given in the question.Then it is not necessarily true take S ,T, to be the set of real numbers and f(x)=x and g(x)=x then gof(x)=x for all x in S which is not constant.

Please rate my answer if it satisfactory. thanks

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