Is $(f(x) + g(x))/2$ equal to $(f(y) + g(y))/2$?

Question

If I have two functions $y = f(x)$ and $y = g(x)$, and I define the mean function as $y = (f(x) + g(x))/2$.

Is this equivalent to rearranging for $x$ and determining $x = (f(y) + g(y))/2$?

Visually, if I want to average two functions graphically (say supply and demand functions) - does my answer change depending on which axis I choose to define the dependent and independent variable?

Answer 1

In other words, we need to determine whether or not the mean of the two functions coincides with the mean of the inverse functions.

Well, assuming the inverse functions exist, this property is not true in general.

Indeed, let consider for example the linear functions:

• $y=x$
• $y=2x$

then we have:

• mean function $y=\frac 3 2 x$
• mean of the inverse functions $ x=\frac 3 4 y$