Solution to 2. Indicate if the following statements are possible. Justify your answer. If the answer is … - Sikademy
Author Image

Archangel Macsika

2. Indicate if the following statements are possible. Justify your answer. If the answer is “yes”, give a specific example of the functions. Let f: and g: Z be two functions. Use a diagram if that helps explain the function. a) Is it possible that f is not onto and g ◦ f is onto? b) Is it possible that g is not onto and g ◦ f is onto? c) Is it possible that g is not one-to-one and g ◦ f is one-to-one?

The Answer to the Question
is below this banner.

Can't find a solution anywhere?

NEED A FAST ANSWER TO ANY QUESTION OR ASSIGNMENT?

Get the Answers Now!

You will get a detailed answer to your question or assignment in the shortest time possible.

Here's the Solution to this Question

Since in question domain and codomain of both functions were not exactly specified, answer will be given in assumption of domain and codomain of both functions being subsets of Z.

a)Yes. Since codomain of g ◦ f function is same as codomain of function g due to definition of function composition. If f(x): A \mapsto B and g(y) : B\mapsto C then g ◦ f(x) : A\mapsto C \ and \ g ◦ f(x)=g(f(x)).

For example if f:Z \mapsto Z f(x)=|x| and g:Z \mapsto Z^+g(x)=|x|. This will result in function g ◦ f(x) : Z\mapsto Z^+ and obviously for every element y in Z+ exist at least one element x in Z such as g ◦ f(x)=y

b)No. As said above function g defines codomain of composition, and if where is not empty subset C' \subset C such as \nexists y\in B \mid g(y) \in C'.

c)Yes. if for example f:Z^+ \mapsto Z^+ f(x)=2x and g:Z \mapsto Z^+ g(x)=|x|. g is not on-to-one function, since exists x1=1 and x2=-1 which make statement g(x_1)=g(x_2) \to x_1=x_2 false. But function g ◦ f is Z^+ \mapsto Z^+function and now for every x1 and x2 g ◦ f(x_1)=g ◦ f(x_2) \to x_1=x_2


Related Answers

Was this answer helpful?

Join our Community to stay in the know

Get updates for similar and other helpful Answers

Question ID: mtid-5-stid-8-sqid-832-qpid-717