Question

How many continuous functions f are there which satisfy the equation `(f(x))^2 = x^2` for all x?

Answer 1

Four

Explanation:

`y^2=x^2` if and only if `y=+-x`.

So the functions `f(x) = x` and `f(x) = -x` two functions that satisfy the equation for all `x`.

Also, the functions that 'mix' these two

`f(x) = absx` `" "` This is `y = {(x,x >=0),(-x,x<0):}`

and `f(x) = -absx` `" "` `y = {(-x,x >=0),(x,x<0):}`