Question

Drew the graph of `y=cos^(-1)sqrt(log_([x])(|x|/x))` where `[*]` represents the greastest integer function.

Answer 1

We have `y=f(x)=cos^(-1)sqrt(log_([x])(|x|/x))`
`|x|/x={{:(1,xgt0),(-1,xlt0):}`
We must have `xgt0`
Also` [x]gt0" and "[x]ne1`
`therefore" "[x]ge2" or "xge2`
From (i) and (ii), the domain of the function is `(2, oo)`
For`xge2, |x|/x=1`
`therefore" "log_([x])(|x|/x)=0-(" for "xge2)`
`therefore" "sqrt(log_([x])(|x|/x))=0(" for "x ge 2)`
`therefore" "cos^(-1)sqrt(log_([x])(|x|/x))=pi/2`
Hence, graph of `y=f(x)" if the line y"=pi/2" for "x in [2, oo)`.