find the dicontinuities of the function. f(x) = (x^2 + 12x + 27)/(x^2 + 4x + 3). there is a removable…

Explanation:

Step1: Factor the numerator and denominator

The numerator $x^{2}+12x + 27=(x + 3)(x+9)$. The denominator $x^{2}+4x + 3=(x + 1)(x + 3)$. So $f(x)=\frac{(x + 3)(x + 9)}{(x + 1)(x + 3)}$.

Step2: Simplify the function

Cancel out the common - factor $(x + 3)$ (for $x\neq - 3$), and we get $f(x)=\frac{x + 9}{x + 1},x\neq - 3$.

Step3: Identify the removable discontinuity

A removable discontinuity occurs when a factor can be canceled out in the rational function. The original function $f(x)=\frac{x^{2}+12x + 27}{x^{2}+4x + 3}$ is undefined at $x=-3$ (because the denominator is 0 at $x=-3$), but the simplified function $\frac{x + 9}{x + 1}$ has a well - defined value at $x=-3$. Substitute $x=-3$ into $\frac{x + 9}{x + 1}$, we have $\frac{-3 + 9}{-3+1}=\frac{6}{-2}=-3$.

Answer:

$(-3,-3)$