How to graph an implicit function by hand?

Question

Suppose we have a function like $\ln(y+1) - y = x$.

What is a strategy for graphing this function without a computer, without a calculator?

Answer 1

Let's sketch $y=\log(x+1)$ first. We know, that $\log(x+1)$ is only defined for $x>-1$ and when $x$ gets closer to $-1$, we know $\log(x+1)$ tends to $-\infty$, and it progresses quite slowly (I assume you know what the graph globally looks like, so that you can sketch it). Also note that $\log(0+1)$ is a zero. Let's also sketch $y=x$, that one is easy. We see:

Let's now take the difference, and we don't want $|\log(x+1)-x|$, but $\log(x+1)-y$, so we sketch how much we go down from the blue line to get to the yellow (and that might be, and will be, negative). We get:

Be sure to include the point $(0,0)$ as that is an "important" point (not necessarily, depending on what you want to do with it, but it's one of the few points that we can actually be sure is on the graph). But wait... We needed $x=\log(y+1)-y$, not $y=\log(x+1)-x$. Oh well, we flip the graph over the line $y=x$, equivalent to flipping over your graph paper and rotating it in the correct position, and we get:

So that's the graph of $x=\log(y+1)-y$ (note that this is not a function).