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Consider the function f(x)={{:(x-[x]-(1)...

Consider the function `f(x)={{:(x-[x]-(1)/(2),x !inI),(0, "x inI):}` where [.] denotes the greatest integer function and I is the set of integers. Then find `g(x)max.[x^(2),f(x),|x|},-2lexle2.`

Text Solution

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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):}`
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