Find the domain of the following function: sin-1 3x2- x - Maths - Inverse Trigonometric Functions

Question

Find the domain of the following function: sin-1
3x2- x

Manbar Singh · Accepted Answer

We have,fx = sin-13x2-xNow, domain of sin-1x = -1, 1So, -1 ≤ 3x2-x ≤ 1CASE 1 : 3x2 - x≥-1⇒3x2 - x + 1 ≥ 0⇒3x - 1232 + 1112≥ 0⇒3x - 1232≥ - 1112This is true for all x∈R CASE 2 : 3x2-x≤1⇒3x2-x-1 ≤0Consider 3x2-x-1 = 0Now, a = 3; b = -1; c = -1Now, D =b2 - 4ac = -12 - 43-1 = 13 >0So, x = -b+D2a and x = -b-D2a⇒x = 1+136 and x = 1-136Now, 3x2-x-1 ≤0⇒x +16-1-13x +16-1+13≤0Now, x +16-1-13 is -ve for x < 1+136Now, x +16-1-13 is +ve for x > 1+136Now, x +16-1+13 is -ve for x < 1-136Now, x +16-1+13 is +ve for x > 1-136⇒1-136 ≤ x ≤ 1+136Combining the above 2 cases, we get1-136 ≤ x ≤ 1+136Domain of sin-13x2-x = 1-136 ≤ x ≤ 1+136