You pick a functional form with some number of parameters. "A logarithmic function" is not a sufficient description. Seeing the vertical asymptote, I would guess th form you want is $y=a \log (3-bx)$ To get the point $(-1,0)$, you need $b=\frac 34$. Now find the $a$ that fits best. If that doesn't satisfy, try another form. If it sounds like a bit of art, it is. You can use root-finders for the parameters once you choose a form.
One way to use transformations to graph $y=\dfrac{1}{\vert x\vert +1}$ is, because of the absolute value, to break it into two parts.
Case I: $x\ge0$
$$ y=\dfrac{1}{x+1} $$
which is the graph of $y=\dfrac{1}{x}$ shifted left by $1$ but we only want the portion of that shifted graph for which $x\ge0$.
Case II: $x<0$
$$ y=\dfrac{1}{-x+1}=-\dfrac{1}{x-1}$$
which is the graph of $y=-\dfrac{1}{x}$ shifted right by $1$ but we only want the portion of that shifted graph for which $x>0$.
Put the two parts together and you have your graph.
This approach becomes quite involved for a graph such as
$$ \vert\,\vert y\vert-3\,\vert=\vert\,3\vert x\vert+2\,\vert$$
because one has to divide the coordinate plane into twelve regions according to whether $y<3,3\le y<0,0\le y<3, y\ge3,x<-\frac{2}{3},-\frac{2}{3}\le x<0,x\ge0 $.
Then one must rewrite the equation without absolute values for each of those sections and graph only the portions of the resulting graphs which fall within those particular sections.
Actually, I see that for this example there are only eight cases because the $x$ expression on the right side of the equation equals $3x+2$ for $x\ge0$ and equals $-3x+2$ for $x<0$.
Here is a desmos.com graph (just the black lines). The orange lines are the graphs of the eight different equations but only the black portions fall within the 'correct' section of the coordinate plane.
Graph of | |y| - 3 | = | 3|x| + 2 |
Best Answer
Set your function equal to a given constant, this give you a function you are used to, and varying the height (ie what you set your function equal to) gives you the graph (2d) of the surface intersected with planes parallel to the xy-plane. Its essentially the same as a contour map of a mountain.