The domain of the function `f(x)=sin^(-1)((8(3)^(x-2))/(1-3^(2(x-1))))` is

Options

A. ` (-oo, 0] `
B. `[2, oo)`
C. `(-oo, 0) uu [2, oo)`
D. `(-oo, -1] uu [1, oo)`

Answer

C

Explanation

f(x) is defined for
`1 le (8.3^(x-2))/( 1-3^(2(x-1))) le 1`
`hArr - 1 le(3^(x) - 3^(x-2))/(1-3^(2x-2)) le 1`
`hArr (3^(x) - 3^(x-2))/(1-3^(2x-2))+ 1 ge 0 and (3^(x)-3^(x-2))/(1-3^(2x - 2)) -1 le 0 `
`hArr (1+3^(x) - 3^(x-2) - 3^(2x-2))/(1-3^(2x-2))ge 0 and (3^(x) - 3^(x-2)-1+3^(2x-2))/(1-3^(2x-2))`
`hArr ((1+3^(x))(1-3^(x-2)))/(3^(x).3^(x-2)) ge and ((3^(x) - 1)(3^(x-2)-1+))/((3^(2x-2) - 1))`
`hArr((3^(x)-3^(2)))/((3^(2x)-3^(2))) ge 0 and ((3^(x) -1))/((3^(2x) - 1)) ge 0`
`hArr ((3^(x) - 3^(2)))/((3^(x) - 3)) ge 0 and ((3^(x) -1))/((3^(x)- 3)) ge 0`
`hArr x in (-oo,1] uu[2,oo) and x in (-oo,0]uu(1,oo)`
`hArr x in (-oo,0] uu [2,oo)`