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

Answer:

$(-3,6)$

Explanation:

Step1: Factor the numerator and denominator

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

Step2: Simplify the function

$f(x)=\frac{x + 9}{x + 1},x\neq - 3$

Step3: Find the value of the simplified - function at the removable - discontinuity point

Substitute $x=-3$ into $\frac{x + 9}{x + 1}$, we get $\frac{-3 + 9}{-3+1}=\frac{6}{-2}=-3$ The original function $f(x)$ has a removable discontinuity at $x=-3$. To find the $y$ - value of the removable discontinuity, we can find the limit as $x\to - 3$ of the original function. Since the simplified function is $\frac{x + 9}{x + 1}$ (for $x\neq - 3$), substituting $x=-3$ into it gives $y = 6$. So the removable discontinuity is at $(-3,6)$.