Let  `f(x) = sqrt(x - 1)/x`  over its implied domain and  `g(x) = text(cosec)^2 x`  for  `0 < x < pi/2`.

The rule for  `f(g(x))`  and the range, respectively, are given by

1. `f(g(x)) = text(cosec)^2(sqrt(x - 1)/x), [1, ∞)`
2. `f(g(x)) = text(cosec)^2(sqrt(x - 1)/x), [2, ∞)`
3. `f(g(x)) = sin(x)cos(x), [−0.5, 0.5]\\ {0}`
4. `f(g(x)) = sin(x)cos(x), (0, 1/2)`
5. `f(g(x)) = 1/2 sin(2x), (0, 1/2]`

The cartesian equation of the relation given by  `x = 3 text(cosec)^2 (t)`  and  `y = 4 cot (t) - 1`  is

A.  `(y + 1)^2/16 - x^2/9 = 1`

B.  `(y + 1)^2 = (16 (x + 3))/3`

C.  `x^2/9 + (y + 1)^2/16 = 1`

D.  `4x - 3y = 15`

E.  `(y + 1)^2 = (16 (x - 3))/3` 

Let  `f(x) = (sqrt(x + 1))/x`  and  `g(x) = tan^2 (x)`, where  `0 < x < pi/2`.

`f(g(x))` is equal to

1. `sin (x) sec^2 (x)`
2. `sec (x) tan^2 (x)`
3. `cos (x) cot^2 (x)`
4. `cos (x) text(cosec)^2 (x)`
5. `text(cosec) (x) cos^2 (x)`