Given function is f(x) = sin-1 x - cos-1(x + 2) + tan-1(x + 1)
∵ Domain of sin-1x and cos-1 x is [-1, 1].
∴ -1 ≤ x ≤ 1 ------(1) [∵ Domain of sin-1 x is (-1, 1).]
And -1 ≤ x+2 ≤ 1 [∵ Domain of cos-1 x is (-1, 1).]
⇒ -1 -2 ≤ x ≤ 1 - 2
⇒ -3 ≤ x ≤ -1 -------(2)
From equation (1) and (2),
we obtain x = -1, (∵ x > -1, x ≤ -1 ⇒ x = -1)
Hence, the domain of f(x) is {-1}.
Now the range of f(x) is
f(-1) = sin-1(-1) - cos-1(-1+2) + tan-1(-1+1)
= -sin-11 - cos-11+tan-1(0)
= \(\frac{-\pi}{2}\) - 0+0 = \(\frac{-\pi}{2}\)
Hence, range of f(x) is \(\{\frac{-\pi}{2}\}\)
Therefore, the number of elements in range of given function f(x) is 1.