Question

Let $f:R\to R,g:R\to R$ be two given fuctions such that $f$ is injective and $g$ is surjective then which of the following is injective

• A. $gof$
• B. $fog$
• C. $gog$
• D. $fof$

Answer 1

The correct option is D $fof$

Since $f$ is injective, for every $x_1,x_2\in R,x_1\ne x_2$ we have $f\left(x_1\right)\ne f\left(x_2\right)$ and hence $f\left(f\left(x_1\right)\right)\ne f\left(f\left(x_2\right)\right)$.

Therefore $f\circ f$ is injective.

Since $g$ is not is injective $g\circ g$, $f\circ g$ and $g\circ f$ are not injective.