Question

let (f) be the function given by (f(x)=\frac{(x - 2)(x - 3)}{(x - 2)(x+1)}). for which of the following values of (x) is (f) not continuous?

a -3 and -1 only

b -3, -1, and 2

c -1 only

d -1 and 2 only

e 2 only

Explanation:

Step1: Recall continuity condition

A rational - function $y = \frac{f(x)}{g(x)}$ is discontinuous where $g(x)=0$.

Step2: Identify the denominator

The function is $f(x)=\frac{(x - 2)(x - 3)}{(x + 3)(x+1)}$, and the denominator is $g(x)=(x + 3)(x + 1)$.

Step3: Find the roots of the denominator

Set $(x + 3)(x + 1)=0$. Using the zero - product property, if $ab = 0$, then $a = 0$ or $b = 0$. So $x+3=0$ gives $x=-3$ and $x + 1=0$ gives $x=-1$.

Answer:

A. -3 and -1 only