Question

If $f:R\to R$ be defined by $f\left(x\right)=e^x$ and $g:R\to R$ be defined by $g\left(x\right)=x^2.$ The mapping $g\circ f:R\to R$ be defined by $(g\circ f)\left(x\right)=g\left[f\right(x\left)\right]\forall x\in R,$ Then

• A. $g\circ f$ is bijective but $f$ is not injective
• B. $g\circ f$ is injective and $g$ is injective
• C. $g\circ f$ is injective but $g$ is not bijective
• D. $g\circ f$ is surjective and $g$ is surjective

Answer 1

The correct option is C $g\circ f$ is injective but $g$ is not bijective

$f\left(x\right)=e^{x}g\left(x\right)=x^{2}g\left(f\right(x\left)\right)=g\left(e^x\right)=(e^x{)}^2=e^{2x}>0\forall x\in R$
Clearly, $g\left(f\right(x\left)\right)$ is injective but $g\left(x\right)$ is not surjective