We have function $g: \mathbb R\to[11/4, \infty)$, with $g(x)=x^2-3x+5$ and we have to check if there exists $h:\mathbb R \to(-1, \infty) $ with property $g \circ h=1_{[11/4, \infty)}$, where $\circ$ means the composition of $g$ and $h$, and if it exists the problem ask to determine it.
I want to say that if $g$ is not a surjective function, that there not exist a inverse to right, but I don't know if it is totally correct, and from there how to find $h$?
Hints:
$g$ is surjective because.
$g(x)=x^2-3x+5=\left(x-\frac{3}{2}\right)^2+\frac{11}{4}$. So the range of $g$ is $\left[\frac{11}{4},\infty\right)$.
Since $g$ is surjective so $g$ does have a right inverse. To find that \begin{align*} y & = x^2-3x+5\\ x^2-3x+(5-y)&=0\\ x & = \frac{3\pm\sqrt{9-4(5-y)}}{2}\\ x & = \frac{3\pm\sqrt{4y-11}}{2}. \end{align*} Now you have to choose the correct branch for the right inverse $h$. Can you proceed from here?
Also you need to define your $h$ separately for $\Bbb{R}-\left[\frac{11}{4},\infty\right)$.