Question

let (f) be the function given by (f(x)=\frac{(x - 2)^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 $y = \frac{g(x)}{h(x)}$ is discontinuous where $h(x)=0$.

Step2: Identify the denominator

The denominator of the function $f(x)=\frac{(x - 2)^2(x + 3)}{(x - 2)(x + 1)}$ is $h(x)=(x - 2)(x + 1)$.

Step3: Solve for discontinuity points

Set $(x - 2)(x + 1)=0$. Using the zero - product property, if $ab = 0$, then $a = 0$ or $b = 0$. So $x-2=0$ gives $x = 2$ and $x + 1=0$ gives $x=-1$.
Although the factor $(x - 2)$ appears in both the numerator and the denominator, the function is still discontinuous at $x = 2$ because the original function is not defined at $x = 2$.

Answer:

D. -1 and 2 only