contraction mapping theorem

I need to show f:[0,1]->[0,1] is a contraction mapping. $\displaystyle f(x)=sinx|x^{0.5}-g(x)|$ where g is continous map form [0,1] to itself

When I try |f(x)-f(y)|, using triangle inequality and sinx<1, I get it is less that or equal to $\displaystyle |x^{0.5}-f(x)|-|y^{0.5}-f(y)|$ which I can't seem to manipulate (even aware that x,f(x) are in [0,1]) to k|x-y| as required.

Re: contraction mapping theorem

I require to show that |f(x)-f(y)| is less that or equal to k|x-y| for some 0<k<1