Chapter 8.3, Problem 4LC

Solution

To find: The points of discontinuity for the rational function $y=\frac{x^2-x-2}{3x^2-7x+2}$ .

The point of discontinuity of the rational function is $x=\frac{1}{3}$ .

Given:

The rational function is $y=\frac{x^2-x-2}{3x^2-7x+2}$ .

Concept used:

Factorize the numerator and denominator of the rational function, cancel the common factors in numerator and denominator and then equate the factors of denominator equal to zero to obtain the points of discontinuity.

Calculation:

Factorize the numerator and denominator of the rational function $y=\frac{x^2-x-2}{3x^2-7x+2}$ . Therefore,

  $\begin{array}{c}y=\frac{x^2-x-2}{3x^2-7x+2}\mathrm{....}\left(\text{Given}\right)\\ y=\frac{\left(x-2\right)\left(x+1\right)}{3x\left(x-2\right)-1\left(x-2\right)}\\ y=\frac{\left(x-2\right)\left(x+1\right)}{\left(3x-1\right)\left(x-2\right)}\\ y=\frac{x+1}{3x-1}\mathrm{....}\left(1\right)\end{array}$

Equating $3x-1=0$ such that $x=\frac{1}{3}$ from $\left(1\right)$ . Therefore, the point of discontinuity of the rational function is $x=\frac{1}{3}$ .

Conclusion:

The point of discontinuity of the rational function is $x=\frac{1}{3}$ .