How can I analytically determine the range of the function?
$$f(x) = \left|\frac{x-2}{x+3}\right|$$
If I have an ordinary linear equation, I'd proceed as follows: \begin{align*} f(x) = x + 2 \Rightarrow y = x + 2 \Rightarrow y - 2 = x \end{align*} then the range of the function is $\mathbb{R}$, but I do not see what to do here.
The result is $[0,\infty)$.
But I want to find out the range numerically, not just estimate it from the graph
I speak about this range. enter image description here
To start with, I'd find the range of the function $g(x)=\frac{x-2}{x+3},$ since $f(x)=\bigl|g(x)\bigr|.$ To do this, first note that $x$ cannot be equal to $-3,$ take an arbitrary $y$ and set $g(x)=y,$ that is, $$\frac{x-2}{x+3}=y,$$ so that you can solve for $x.$ The first step in this process is to note that since $x\neq -3,$ then $$x-2=(x-3)y.$$ By the end, you'll have an equation of the form $$x=\frac{ay+b}{cy+d}.$$ From this, you'll be able to find the one value that $y$ cannot take in the equation $g(x)=y,$ which will tell you the range of $g,$ from which you'll conclude the range of $f.$