Find a formula for the single variable function $f(0,y)$

Question

If I place $x=0$, then $f(0,y)=0$. If I disregard $x$ then $f=\frac8y$

I can't see any one of these being correct.

Answer 1

Just as you say, $f(0,y) = 0$. I was going to add the caveat that $f(0,0)$ isn't defined, but in fact it looks like $f(x,y)$ is meant to have a piecewise definition, with $f(0,0)$ explicitly defined as 0, so there isn't even that issue.

As noted in comments, your second notion (of "disregarding $x$") doesn't make sense: you evaluate $f(0,y)$ by replacing $x$ with $0$.

Answer 2

If I place x=0, then f(0,y)=0.

This indeed seems to be the case - in the image, the z-value (which I assume corresponds to $f$) is 0 on the x-axis.

If I disregard x then f=8y

What does "disregarding $x$" mean? You can't just imagine $x$ is not there - where exactly does the function behave like $\frac{8}{y}$? I assume you came there through $\frac{8y}{y^2}$, but to make the numerator $8y$, you need $x = 1$, then the denominator becomes $y^2 + 1$.