Question

Can we apply inequality rules for operations to functions? If yes, under what conditions? Are there counter examples?

Answer 1

Yes, we can apply. If you apply a function f(x) to the statement a<b, you'd typically like to conclude something like f(a)<f(b) or f(a)>f(b). In other words, you want to know whether the process preserves the direction of inequality or reverses it.

But to draw such a conclusion, you generally need to know whether f(x) is

• increasing, i.e., a<b⟺f(a)<f(b) or
• decreasing, i.e., a<b⟺f(a)>f(b)

on the interval from a to b.

For example: f(x)=x+2 is always increasing, and g(x)=−x is always decreasing, so applying f to a<b yields

a+2=f(a)<f(b)=b+2a+2=f(a)<f(b)=b+2

but applying g yields

−a=g(a)>g(b)=−b