Question

at what points is the function y = \frac{x + 2}{x^{2}-9x + 14} continuous?
describe the set of x - values where the function is continuous, using interval notation.
(simplify your answer. type your answer in interval notation.)

Explanation:

Step1: Factor the denominator

Factor $x^{2}-9x + 14$ as $(x - 2)(x - 7)$. So the function is $y=\frac{x + 2}{(x - 2)(x - 7)}$.

Step2: Find the points of discontinuity

A rational - function is discontinuous where the denominator is zero. Set $(x - 2)(x - 7)=0$. Solving $x-2 = 0$ gives $x = 2$, and solving $x - 7=0$ gives $x = 7$.

Step3: Determine the intervals of continuity

The function is continuous for all real numbers except $x = 2$ and $x = 7$. In interval notation, the set of $x$ - values for which the function is continuous is $(-\infty,2)\cup(2,7)\cup(7,\infty)$.

Answer:

$(-\infty,2)\cup(2,7)\cup(7,\infty)$