The number of integral values of a for which the equation ⁻¹ ( 1-x^2 1+x^2 ) - ⁻¹x = a has exactly one real root is

Step-by-step solution

Let f(x) = ⁻¹ ( 1-x^2 1+x^2 ) - ⁻¹x . We know the piecewise definition for the inverse cosine term: ⁻¹ ( 1-x^2 1+x^2 ) = cases 2 ⁻¹x, & x 0 -2 ⁻¹x, & x Thus, the function f(x) can be written as: f(x) = cases 2 ⁻¹x - ⁻¹x = ⁻¹x, & x 0 -2 ⁻¹x - ⁻¹x = -3 ⁻¹x, & x Let us analyze the range of f(x) in both intervals: For x 0 , ⁻¹x [0, 2 ) . The function strictly increases from 0 to 2 . For x To have exactly one real root, the horizontal line y = a must intersect the graph of f(x) exactly once. From the ranges above: - For