Calculate the range of a rational expresstion in x, where x is real:

STEP 1:   Put the provided rational expression same to y and form the quadratic equation in x.

STEP 2:   Calculate the discriminant D of the quadratic equation calculated in step 1.

STEP 3:   Since x is real, Thus, put D ≥ 0. We obtain an inequation in y.

STEP 4:   Solve the above inequation for y. The range of y so obtained calculates the range attained by the shown rational expression.

Illustration: Calculate the maximum and minimum values of  for real values of x.

Solution:                   Suppose 

                             =>(y - 1)x² + (y + 1)x + (y - 1) = 0. On solving for x, we obtain

                                    Now for real values of x we can get real values of y, therefore x to be real

                                    .

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