Question

Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but g is not injective. (Hint: Consider f(x) = x and g(x) = |x|)

Answer 1

example -1 : Let $f(x)= x , g(x) = |x|$
here consider g(x) = |x| => g is not injective mapping.
since, -1 and 1 have the same image 1.
but gof = g(f(x)) = g(x) = |x|
here f : N → R and g : R → R then, gof : N → R
e.g., domain of gof belongs to N .
hence, for every natural number n ,g(n) = |n| = n , gof has unique image .
so, gof is injective or one one .

example - 2 :- Let f : N → R be defined as f(x) = 2x and g : R → R be defined as g(x) = x².
here you can also observed
gof : N → R be defined as gof = 4x² is injective while g(x) is not injective.