The function $f:\left(-\infty ,-1\right]\to \left(0,e^5\right]$ defined by $f\left(x\right)=e^{x^3-3x+2}$ is

$\begin{array}{l}f\left(x\right)=e^{x^3-3x+2}\\ \mathrm{Let}g\left(x\right)=x^3-3x+2;\\ g\text{'}\left(x\right)=3x^2-3=\left(x^2-1\right)\\ \ge 0\mathrm{for}x\in \left(-\infty ,-1\right]\end{array}$

Therefore, $f\left(x\right)$ is increasing function. Hence $f\left(x\right)$ is one-one.

Now, the rang e of  $f\left(x\right)$ is $\left(0,e^4\right]$.

But co-domain is$\left(0,e^5\right]$

Hence, $f\left(x\right)$ is an into function.