Let f: B → C and g: A → B be two functions and let h = f o g. Given that h is…
2005
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
Attempted by 191 students.
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Correct answer: B
Answer: f should be onto but g need not be onto.
Reasoning:
If f ∘ g is onto, then for every element c in C there exists some a in A with f(g(a)) = c.
Let b = g(a) (an element of B). Then f(b) = c, so for every c in C there is some b in B with f(b) = c. Thus f is onto.
There is no requirement that g hit every element of B. It only needs to hit enough elements so that f maps those to all of C.
Counterexample showing g need not be onto:
Take A = {a}, B = {b1, b2}, C = {c}.
Define g(a) = b1, and define f(b1) = c and f(b2) = c.
Then f is onto C, f ∘ g maps a to c (so f ∘ g is onto), but g is not onto B because b2 is not in the image of g.
Therefore the correct conclusion is that f must be onto, while g need not be onto.