Given, $f\left(x\right)=e^{x^3-3x+2}$
Let, $g\left(x\right)=x^3-3x+2$
$\Rightarrow g^{\prime }\left(x\right)=3x^2-3=3\left(x^2-1\right)$
$\Rightarrow g^{\text{'}}\left(x\right)\ge 0, \text{for }x\in \left(-\infty ,-1\right]$ and 

$g^{\text{'}}\left(x\right)\le 0\text{\hspace{0.17em}for\hspace{0.17em}}x\in \left[-1,1\right]$
$\therefore g\left(x\right)$ is increasing as well as decreasing in different intervals
$\therefore f\left(x\right)$ is many-one
Now, Range of $f\left(x\right)$ is $\left(0,e^4\right],$ but co-domain is $\left(0,e^5\right].$
$\therefore f\left(x\right)$ is into function