Consider a function f(x) =

A function of this type may be discontinuous for some values of x because the denominator may tend to zero as x approaches to one of the roots (if any) of px 2 + qx + r = 0. There may, too, be limitations on the values which the function may take which follow from the relations between the coefficients when y = is expressed in theform f(x, y) = 0 and arranged as a quadratic equation in x. For example we discuss the function f(x) =

Let y = f(x) = ⇒ x 2 y –x –y = 0

Since x can have any value, so, 1 + 4y 2 > 0

This inequality holds for all values of y and therefore y can have any value. That is – ∞ < f(x) < ∞ . Again the function is discontinuous at x = –1 and x = 1 As x → 1 + 0, y → + ∞ , since each of three factors x, and is positive for these values of x. Similarly as x → 1 – 0 y → – ∞ as x → –1 + 0 y → + ∞ as x → –1 – 0 y → – ∞

That is, x = 1 and x = –1 are vertical asymptotes to the curve of the function.

Since, y = , as x → + ∞ , y → 0 + 0 and as x → ∞ , y → 0 – 0

That is y = 0 is horizontal asymptote.

Further, = = – < 0

∴ y has no stationary values and it decreases for all values of x. The graph of y = f(x) is shown in the above fig.

(i) The function y = has-