Let `f(x)=sgn(cot^(-1)x)+tan(pi/2[x]),` where `[x]` is the greatest integer function less than or equal to `x ,` then which of the following alternatives is/are true? `f(x)` is many-one but not an even function. `f(x)` is a periodic function. `f(x)` is a bounded function. The graph of `f(x)` remains above the x-axis.

Options

A. `f(x)` is many-one but not an even function.
B. `f(x)` is a periodic function.
C. `f(x)` is a bounded function.
D. The graph of `f(x)` remains above the x-axis.

Answer

A, B, C, D

Explanation

`f(x)=sgn (cot^(-1)x)+tan((pi)/(2)[x])`
`sgn(cot^(-1)x)` is dedfined when `cot^(-1)x` is defined, which is `AA x in R.`
`tan((pi)/(2)[x])` is defined when `(pi)/(2)[x] ne ((2n+1))/(2)pi, " where " n in Z,`
`or [x] ne 2n+1 or x in [2n+1, 2n+2).`
Hence, domain of `f(x) " is " underset (n in Z)(cup) [2n, 2n+1).`
Also, `cot^(-1)x gt 0 AA x in R.` Then
`f(x)=1+tan ((pi)/(2)[x])=1`
` :. f(x)=1, x in D_(f)`

From the graph, `f(x)` si periodic with period 2.