Question

Domain of the function f(x) = sin^-1 (1 + 3x + 2x²) is ______.

Options

• (- ∞, ∞)

• (- 1, 1)

• `[- 3/2, 0]`

• `(- ∞, (- 1)/2) ∪ (2, ∞)`

Answer 1

Domain of the function f(x) = sin^-1 (1 + 3x + 2x²) is `[- 3/2, 0]`.

Explanation:

- 1 ≤ 1 + 3x + 2x² ≤ 1

Case I: 2x² + 3x + 1 ≥ - 1;  2x² + 3x + 1 ≥ 0

x = `(- 3 +- sqrt(9 - 16))/4 = (- 3 +- "i"sqrt7)/4` (imagmary).

Case II: 2x² + 3x + 1 ≤ 1

`=> 2x^2 + 3x <= 0 => 2x(x + 3/2) <= 0`

`=> (- 3)/2 <= x <= 0 => x ∈ [- 3/2, 0]`

In case I, we get imaginary value hence, rejected

∴ Domain of function = `[(-3)/2, 0]`