How to find the graphs of the following multi variable functions: $f(x,y)=ln(1-xy)$ and $f(x,y)=\frac{y}{e^x}$ As the title says, how would you mathematically approach these problems? These are problems on a sample exam, but the answers do not have explanations as to how they derived the answers. 


 A: A great way of approaching 3d space in multi-variable calculus is to limit yourself to two dimensions. What I mean by this is to hold one variable constant while varying the other thus giving you a cross-sectional view of the 3d representation of the function.  
Problem #1) Analyze the graphs f(x) = $\frac{1}{e^x}$ = $e^{-x}$ (letting y = 1) giving you a cross-section within the xz-plane and g(y) = y (letting $e^{-x}$ = 1), a cross-section within the yz-plane.  
See f(x) here:

See g(y) here:

Using these cross-sections of the graphs, I hope that you can see that the correct answer to this question is the top left graph (the first a) as the sign of y reflects $e^{-x}$ over the x-axis for y $\lt 0$. Additionally, the graph appears to flatten out as x increases because the denominator increases at an exponential rate causing f(x,y) to appear flat (but you can definitely notice the z = y cross-section when $x \lt 0$.  
Problem #2) When analyzing the domain of a multi-variable function, you must first notice the constraints of the problem. For example, $y = \frac{1}{x}$ has the constraint that the denominator must not be zero, therefore $x \ne 0$. Your problem has the constraint:
$$1 - xy \gt 0 \Rightarrow xy \lt 1 \Rightarrow y \lt \frac{1}{x}$$
As you can see the problem gives you this exact curve. Therefore the answer to this question is c) Regions III, IV, V, VI not including the curve.Hope this helps!
