I have a function: $f(x,y) = y^2 + \ln(x^2) -1$.
I need to sketch this function and use the sketch to obtain the domain of the function, however, I cannot sketch the function if it is in the form given above, I need to simplify/rearrange it into to something I can sketch.
This is where I am having some issues. I'm not sure how I can simplify the above into something I can sketch, the only simplification I can see possible is to change $\ln(x^2)$ to $2\ln(|x|)$ but I don't think this helps.
Any suggestions on what simplification I could make or what process I would need to do to be able to sketch this function?
Thanks in advance
I'd say in this case it is easier not to sketch this function, but to look at this algebraically. The function $g(y)=y^2$ is defined for all $y \in \mathbb{R}$. The function $h(x)=\ln(x^2)$ is well-defined if $x^2>0$, i.e. if $x \neq 0$. Since $f(x,y) = g(y)+h(x)-1$, your function is well-defined for all $(x,y) \in (\mathbb{R}\setminus\{0\}) \times \mathbb{R}$, so that is your domain.