I am trying to learn how to work with functions and I have some things that I didn't fully understand. How do I multiply and plot a function that is the result of a multiplication of two other subfunctions with two variables f(x,y)?
I have started with a simple random example, as explained below:
I consider two functions with one variable. I can easily calculate a product between them and get a plot that represents the final result. For $f(x) = x^2$, $g(x) = \sqrt{x} $ and $h(x) = f(x)g(x)$ I get a nice and smooth final function, which I can plot (using Octave) and calculate manually to validate it (see this plot). I can also get the area under the curve and everything works as I expected. It's also easy to build plots with x on the x-axis and f(x),g(x) and h(x) on the y-axis.
The challenge now is how to do this when there are two functions with two variables? For example, if I have $f(x) = x^2$, $g(y) = \sqrt{y}$, how do I calculate $h(x,y)=f(x)g(y)$? It can't be simply $h(x,y) = x^2 \sqrt{y}$? I'm trying to build the plot manually and I fail to understand where to place the functions. Can anyone help me please to understand how this works and how to build the plot?
When you graph a function $y = f(x)$ of one variable you use the $x$-axis for the independent variable and the $y$ axis for the value of the function. The graph is a curve.
When you graph a function $z= h(x,y)$ of two variable you use the whole $x$-$y$ plane for the independent variables and the $z$ axis for the value of the function. The graph is a surface. On that surface yhou can see curves corresponding to the values of, say, $h(c,y)$ where you keep $x=c$ fixed and watch what happens as $y$ varies.
Here's a picture for your example:
drawn by desmos: https://www.desmos.com/calculator/nqom2ih05g
You can see the parabolas corresponding to $h(x,c) = \sqrt{c}x^2$ for various values of $c$.