Operations on Functions and How it Changes Their Graphs

Question

This might sound like a dumb question, but it’s something I’ve been thinking of for a long time. Say you have two functions $f(x)$ and $g(x)$. We can graph them and analyse the graphs easily. Now, using only the graphs of $f(x)$ and $g(x)$, can you deduce what the graph of $f(x)+g(x)$ might look like? What about $f(x)-g(x)$? What about any other operation that we may perform on the functions (multiplication, division, exponentiation, etc.)?
1.Here is the graph of $y=x^2$.
2.Here is the graph of $y=e^x$.
3. Here is the graph of $y=e^x+x^2$.
I simply don’t see how one can get from graph $1$ and graph $2$ to graph “$1+2$”. That’s just an example, though. I’m looking for a more general insight as stated above.

Answer 1

So this might be an interesting problem to consider.

Is true that for any graph of your choice of the form $G(x,y)=0$

There exists a coordinate system $u,v$ such that the sum of $x^2$ and $e^x$ in this coordinate system equals that graph?

What would this entail to prove? We want some functions $X,Y$ such that

$$ x = X(u,v)\\ y = Y(u, v)$$

So then we can rewrite

$$ y = x^2 \rightarrow Y(u,v) = X(u,v)^2$$ $$ y = e^x \rightarrow Y(u,v) = e^{X(u,v)}$$

And we should be able to rewrite these equations as

$$ u = F_1(v) $$ $$ u = F_2(v) $$

Such that the graph

$$ u = F_1(v) + F_2(v)$$

Should be identical to the graph $G(x,y) = 0$.

This would be a pretty tricky theorem to prove but personally my hunch is that it is always possible regardless of the choice of $G$.

Answer 2

As I am not yet able to write comments, please excuse that I use an answer for what should be a comment.

What you can always do is, of course, plot both graphs and then "add" the height of one graph onto the other one, but I'd consider it cheating in this case.

Generally, what you are asking for is tricky. You might want to try to do that for two linear maps f(x)=ax and g(x)=bx first. Even then, you will notice that it is not trivial to figure out the slope of f(x)+g(x) = (a+b)x without some arithmetics. Naively, one might assume that it suffices to plot f, and then rotate according to the slope of g. But already for the case of f(x)=x=g(x), such a strategy would leave you with something that is not a function.

It is a really interesting question, nonetheless, and someone else might have some more tricks up their sleeves.