Answer
Domain: all real numbers, $q\neq 5 / 4$
Range: all real numbers, $s^{-1}(y)\neq -1/2$
Work Step by Step
Since division by 0 is undefined, $s(q)$ is not defined when $5-4 q=0$. So the domain of $s(q)$ is all real numbers except $q=5 / 4$.
To find the range we find the inverse function. We solve the equation $y=s(q)=\frac{2 q+3}{5-4 q}$ for $q$.
$$
\begin{aligned}
y & =\frac{2 q+3}{5-4 q} \\
y(5-4 q) & =2 q+3 \\
5 y-4 q y & =2 q+3 \\
5 y-3 & =q(2+4 y) \\
q & =\frac{5 y-3}{2+4 y} \\
s^{-1}(y) & =\frac{5 y-3}{2+4 y}
\end{aligned}
$$
The domain of the inverse function is all real numbers except $-1 / 2$, so the range of $s(q)$ is all real numbers except $-1 / 2$.
