Let f: B → C and g: A → B be two functions and let h = f o g. Given that h is an onto function. Which one of the following is TRUE?
(A) f and g should both be onto functions.
(B) f should be onto but g need not be onto
(C) g should be onto but f need not be onto
(D) both f and g need not be onto
Answer: (B)
Explanation: A function f: X → Y is called on-to function if for every value in set Y, there is a value in set X.
Given that, f: B → C and g: A → B and h = f o g. Note that the sign o represents composition. h is basically f(g(x)). So h is a function from set A to set C. It is also given that h is an onto function which means for every value in C there is a value in A.
We map from C to A using B. So for every value in C, there must be a value in B. It means f must be onto.
But g may or may not be onto as there may be some values in B which don’t map to A.
Example :
Let us consider following sets A : {a1, a2, a3} B : {b1, b2} C : {c1} And following function values f(b1) = c1 g(a1) = b1, g(a2) = b1, g(a3) = b1 Values of h() would be, h(a1) = c1, h(a2) = c1, h(a3) = c1 Here h is onto, therefore f is onto, but g is onto as b2 is not mapped to any value in A.
Given that, f: B → C and g: A → B and h = f o g.