Does f(3x) = 3f(x)?

Question

Is just listed as a true or false question in my book but I was just thinking when this actually is true or false. For instance if f(x) = cx for some c then this equality hold. Or when f = 0 for all x. Am I right?

Answer 1

Of course, there exist functions $f$ which satisfy this equation for all $x$, which you have found. When $f(x) = cx$ for some $c$ (including $c=0$), the equation will hold true.

The equation is what's called a functional equation, which is an equation where you're trying to solve for a function rather a variable. Existence of solutions is guaranteed for some certain kinds of functional equations.

Alternatively, you could look for $x$ such that the equation holds true for all $f$, which has no solutions in this case, or for a given $f$, which would simply be solving the equation.

However, for the purpose of the question in the book, the answer is false, as the statement is not true for all $f$ and all $x$, which is the requirement for it to be true if no additional conditions/restrictions are listed.

Answer 2

Just note that when $f(x)=x^2$, $$f(3x)=9f(x)$$