Let f be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a in A. Which of the following statements is always true for all such functions f and g? A. g is onto => h is onto B. h is onto => f is onto C. h is onto => g is onto D. h is onto => f and g are onto

GATE 2005 · Discrete Mathematics · Functions · medium

Answer: h is onto => g is onto (Option C)

1. Show h onto forces g onto: h onto means every c in C equals h(a) = g(f(a)) for some a, so c = g(b) with b = f(a) in B; thus every c in C is an image under g, i.e. g is onto
2. Rule out the other options by counterexample: take A={1}, B={x,y}, C={c} with f(1)=x and g(x)=g(y)=c; then h(1)=c is onto C, yet f misses y so f is not onto, killing B and D; option A fails since g onto does not force h to reach C when f's image is too small