The range of the function `y=[x^2]-[x]^2` `x in [0,2]` (where [] denotes the greatest integer function), is

Options

A. [0]
B. [0,1]
C. [1,2]
D. [0,1,2]

Answer

D

Explanation

We have,
`{:(,y=[x^(2)]-[x]^(2)",",x in[0,2]),("i.e.,",y=[x^(2)]",",0lexlt1),(,y=[x^(2)]-1",",1lexlt2),(,y=0,x=2),("i.e.,",y=0",",0lexlt1),(,=1-1=0",",1lexltsqrt2),(,=2-1=1",",sqrt2lexltsqrt3),(,=3-1=2",",sqrt2lexltsqrt3),(,=3-1=2",",sqrt3lexlt2),(,=0,x=2):}`
Hence, the range is `{0,1,2}.`