The curve y = ⁻¹ x + ⁻¹(x+1) and the horizontal line y = ⁻¹ 2 intersect at a unique point (x₀, y₀) . The value of x₀^2 + x₀ is :

Step-by-step solution

To find the intersection point, equate the two expressions: ⁻¹ x + ⁻¹(x+1) = ⁻¹ 2 Assume x > -1 , which implies x+1 > 0 . Thus, ⁻¹(x+1) = ⁻¹ ( 1 x+1 ) . The equation becomes: ⁻¹ x + ⁻¹ ( 1 x+1 ) = ⁻¹ 2 Check the product of the arguments: x 1 x+1 = x x+1 Since x > -1 , x x+1 ⁻¹ ( x + 1 x+1 1 - x x+1 ) = ⁻¹ 2 Simplify the argument: x^2+x+1 x+1 x+1-x x+1 = x^2+x+1 So, ⁻¹(x^2+x+1) = ⁻¹ 2 x^2+x+1 = 2 x^2+x = 1 The roots are x = -1 5 2 . The positive root satisfies x > -1 . For x Since x₀ is the unique solution, it satis