I have a function
$h\left(x\right)=\sqrt{\mu x + \eta}$ such that $\mu\in\left(0,+\infty\right)$ and $\eta\in\mathbb{R}\setminus\left\{0\right\}$.
I want to check the domain and range of $h$ in case the coefficient $\eta<0$ or $\eta>0$.
To do so, I proceeded as$\colon$
$h\left(x\right)=\begin{cases} \sqrt{\mu x + \eta}, \hspace{20mm} \eta>0\\ \sqrt{\mu x - \eta}, \hspace{20mm} \eta<0\\ \end{cases} $
and concluded that the domain of $h$, namely $D_h$ is found as
$D_h=\begin{cases} x\in\left[\frac{-\eta}{\mu},+\infty\right), \hspace{20mm} \text{since}\hspace{2mm}\sqrt{\mu x + \eta}>0 \\ x\in\left[\frac{+\eta}{\mu},+\infty\right), \hspace{20mm} \text{since}\hspace{2mm}\sqrt{\mu x - \eta}>0 \end{cases} $
Also, I found the range of $h$, namely $R_h=\left[0,+\infty\right)$ in both above cases.
Now, I want to observe that under what conditions $h$ is a contraction mapping, namely $R_h\subset D_h$.
In case $\eta<0$, then I need to have
$\underbrace{\left[0,\infty\right)}_{R_h} \subset \underbrace{\left[\frac{+\eta}{\mu},+\infty\right)}_{D_h} \implies \frac{\eta}{\mu}<0 \hspace{5mm} \implies \hspace{5mm} \eta<0 \hspace{5mm} \text{But I already have this satisfied.} $
And In case $\eta>0$, then I need to have
$\underbrace{\left[0,\infty\right)}_{R_h}\subset \underbrace{\left[\frac{-\eta}{\mu},+\infty\right)}_{D_h} \implies \frac{-\eta}{\mu}<0 \hspace{5mm} \implies \hspace{5mm} \eta>0 \hspace{5mm} \text{But I already have this satisfied.} $
Is this approach correct?