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

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Answer 1 · 2016-12-09T18:56:19

Four

$y^{2} = x^{2}$ $y^2=x^2$ if and only if $y = \pm x$ $y=+-x$ .

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

Also, the functions that 'mix' these two

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

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

How many continuous functions f are there which satisfy the equation (f(x))^2 = x^2(f(x))2=x2(f(x))^2 = x^2 for all x?