Question

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 $h\left(a\right)=g\left(f\right(a\left)\right)$ for all $aϵA$. Which of the following statements is always true for all such functions $f$ and $g$ ?

• A. $g$ is onto $\Rightarrow$ $h$ is onto
• B. $h$ is onto $\Rightarrow$ $f$ is onto
• C. $h$ is onto $\Rightarrow$ $g$ is onto
• D. $h$ is onto $\Rightarrow$ $f$ and $g$ are onto

Answer 1

The correct option is C $h$ is onto $\Rightarrow$ $g$ is onto

Given, $h=g\left(f\right(x\left)\right)=g.f$
Consider the following arrow diagram:

From above diagram it is clear that
$g$ is not onto $\Rightarrow h=g.f$ is also not onto, since the co-domain of $g$ is same as the co-domain of $g.f.$
The contrapositive version of the above implication is
$h$ is onto $\Rightarrow g$ is onto
which also has to be true since direct $\equiv$ contrapositive.
So option (c) is true.