If the function g o f is defined and onto, then

Question

If the function g o f is defined and onto, then (A) neither f nor g is onto (B) f and g both are necessarily onto (C) f must be onto (D) None of the above

Tardigrade Admin · Accepted Answer

Consider f: 1,2,3,4 arrow 1,2,3,4 and g: 1,2,3,4 arrow 1,2,3 defined as f(1)=1, f(2)=2, f(3)=f(4)=3, g(1)=1, g(2)=2 and g(3)=g(4)=3. It can be seen that gof is onto but f is not onto.

Q. If the function $g o f$ is defined and onto, then

neither $f$ nor $g$ is onto

$f$ and $g$ both are necessarily onto

$f$ must be onto

None of the above

Consider $f : {1, 2, 3, 4} \to {1, 2, 3, 4}$ and $g : {1, 2, 3, 4} \to {1, 2, 3}$ defined as $f (1) = 1, f (2) = 2$ , $f (3) = f (4) = 3, g (1) = 1, g (2) = 2$ and $g (3) = g (4) = 3$ . It can be seen that gof is onto but $f$ is not onto.

Q. If the function gof is defined and onto, then

neither f nor g is onto

f and g both are necessarily onto

f must be onto