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Assumed I want to draw three simple graphs based on the following illustration:

How can I plot those three graphs? I've tried to create something like this:

Minimum Working Example (MWE):

documentclassstandalone
usepackagepgfplots

begindocument
 begintikzpicture
 beginaxis[domain=0:1]
 addplot+[no marks, samples=100] exp(3*x);
 addplot+[no marks, samples=100] (-exp(3*x)+20);
 endaxis
 endtikzpicture
enddocument

Screenshot of the current state:

Description of the issue:

However, I don't get the point on how to figure out the function of the third graph as well as how to clip the graphs in x and y direction to align them centered in the plot. It is not necessary to be completely precise, the accuracy of my upper template will be fine enough.

How to do that?

You can use the function f1*f2/(f1+f2) to get something like what you describe. To make the code a bit clearer I defined the functions f1 and f2.

documentclassstandalone
usepackagepgfplots
pgfplotsset%
 compat=1.12,
 /pgf/declare function=f1(x)=exp(3*x);,
 /pgf/declare function=f2(x)=-exp(3*x)+20;,

begindocument
 begintikzpicture
 beginaxis[domain=0:1]
 addplot+[no marks, samples=100] f1(x);
 addplot+[no marks, samples=100] f2(x);
 addplot+[no marks, samples=100] f1(x)*f2(x)/(f1(x)+f2(x));
 endaxis
 endtikzpicture
enddocument

If you want the function to follow more closely you can use sqrt(f1^2*f2^2/(f1^2+f2^2). Then replace the third plot with

addplot+[no marks, samples=100] sqrt(f1(x)*f1(x)*f2(x)*f2(x)/(f1(x)*f1(x)+f2(x)*f2(x)));

to get

Are you looking for the minimum? (I shifted the plot a bit but you can of course undo the shift.)

documentclass[border=3.14mm,tikz]standalone
usepackagepgfplots
pgfplotssetcompat=1.16
begindocument
 begintikzpicture
 beginaxis[domain=0:1]
 addplot+[no marks, samples=100] exp(3*x);
 addplot+[no marks, samples=100] (-exp(3*x)+20);
 addplot+[no marks, samples=6,smooth]
 min(exp(3*x),(-exp(3*x)+20))-pi/2;
 endaxis
 endtikzpicture
enddocument