# How to multiply two functions with two variables and manually build a plot

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?

• Welcome to stackexchange. It's just as simple as you say it is. In the plot the graph over each line with fixed $x$ is just a multiple of the graph of $g$, and vice versa. With that information you should be able to sketch the particular example you give. Feb 22, 2018 at 13:11
• Thank you. So if I have x = 1 and y = 1, I keep them on the x-axis and y-axis and the function resulted from their multiplication will be on the z-axis or in some kind of projection between the two coordinates made by x-axis and y-axis, e.g. 1,1? Is it that simple and correct? Feb 22, 2018 at 13:13
• You should remove the multiplicative-function tag. It doesn't mean what you think it means. Apr 11, 2018 at 21:08

## 1 Answer

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 $c$.