Consider the function `f(x)={x-[x]-1/2,ifx !in Io,ifx in I ,w h e r e[dot]` denotes the fractional integral function and `I` is the set of integers. Then find `g(x)=m a xdot{x^2,f(x),|x|;-2lt=xlt=2.`

Explanation

We first draw the graphs of `y=f(x),y=x^(2)andy=|x|``y=f(x),y=x^(2)andy=|x|`.
To deaw the graph of `y=x-[x]-0.5={x}-0.5`, we shift the graph of `y={x},0.5` units downward vertically for non-nitegral values of x and plotting '0' for integral values fo x as shown in the following in the following figure.

Clearly, from the graph, g(.x) =`{{:(x^(2)",",-2lexle-1),(1-x",",-1lexle-1//4),((1)/(2)+x",",-(1)/(4)ltxlt0),(1+x",",0lexle1),(x^(2)",",1lele2):}`