Domain of the function f(x)=`sin^(- 1)(1+3x+2x^2)`

Explanation

`f(x) = sin^(-1)(1+3x+2x^(2))`
We must have ` -1 le 1 +3x +2x^(2) le 1`
Now `2x^(2)+3x +1 ge -1`
`implies 2x^(2)+3x +2 ge 0`, which is always true as discriminant ` le 0`.
Also, `2x^(2) +3x +1 le 1`
`implies 2x^(2) +3x le 0`
`implies x(2x+3) le 0`
`implies (-3)/(2) le x le 0`
`implies x in [-(3)/(2),0]`