If for real values of x, =x+ 1 x , then

Step-by-step solution

Given the equation = x + 1 x for real values of x . For any real x 0 , the expression x + 1 x satisfies the inequality |x + 1 x | 2 by the Arithmetic Mean-Geometric Mean inequality (AM-GM) for positive x , and |x + 1 x | -2 for negative x . Specifically, if x > 0 , then x + 1 x 2 . If x 0 , then x + 1 x = -(y + 1 y ) -2 . Thus, the range of the function f(x) = x + 1 x is (- , -2] [2, ) . The range of is [-1, 1] . Since the range of x + 1 x and the range of have no common values except possibly at the boundaries, we