The total number of real solutions to the equation ⁻¹ ( 2x 1-x^2 ) + ⁻¹ ( 1-x^2 2x ) = 2 3 is _____ .

Step-by-step solution

Let f(x) = ⁻¹ ( 2x 1-x^2 ) and g(x) = ⁻¹ ( 1-x^2 2x ) . We use the piecewise definitions of f(x) : f(x) = 2 ⁻¹x for x (-1, 1) f(x) = 2 ⁻¹x - for x > 1 f(x) = 2 ⁻¹x + for x For g(x) , we use ⁻¹(z) = ⁻¹ ( 1 z ) if z > 0 , and + ⁻¹ ( 1 z ) if z Here z = 1-x^2 2x . z > 0 x^2-1 2x z We analyze the equation f(x) + g(x) = 2 3 in four intervals: Interval 1: x (0, 1) Here z > 0 , so g(x) = ⁻¹ ( 2x 1-x^2 ) = f(x) . LHS = 2f(x) = 4 ⁻¹x . 4 ⁻¹x = 2 3 ⁻¹x = 6 x = 1 3 . Since 1 3 (0, 1) , this is 1 valid solution. Interval 2: x