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

Dec 9, 2016

Four

#### Explanation:

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

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

Also, the functions that 'mix' these two

$f \left(x\right) = \left\mid x \right\mid$ $\text{ }$ This is $y = \left\{\begin{matrix}x & x \ge 0 \\ - x & x < 0\end{matrix}\right.$

and $f \left(x\right) = - \left\mid x \right\mid$ $\text{ }$ $y = \left\{\begin{matrix}- x & x \ge 0 \\ x & x < 0\end{matrix}\right.$