find the dicontinuities of the function.

$f(x) = \frac{x^2 + 12x + 27}{x^2 + 4x + 3}$.

there is a removable discontinuity at
$(\quad, \quad)$.

Question

find the dicontinuities of the function.

$f(x) = \frac{x^2 + 12x + 27}{x^2 + 4x + 3}$.

there is a removable discontinuity at
$(\quad, \quad)$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Polynomial Factoring", "Holes in Rational Graphs", "Removable Discontinuities" ], "new_concepts": [], "current_concepts": [ "Rational Functions", "Polynomial Factoring", "Holes in Rational Graphs", "Removable Discontinuities" ] } ## Factor the numerator and denominator Using the *Polynomial Factoring* knowledge point $$ \begin{aligned} x^2 + 12x + 27 &= (x + 3)(x + 9) \ x^2 + 4x + 3 &= (x + 3)(x + 1) \end{aligned} $$ ## Identify the removable discontinuity Using the *Removable Discontinuities* and *Holes in Rational Graphs* knowledge points $$ \begin{aligned} &f(x) = \frac{(x + 3)(x + 9)}{(x + 3)(x + 1)} \ &\text{Common factor: } x + 3 = 0 \implies x = -3 \end{aligned} $$ ## Calculate the y-coordinate of the hole Using the *Holes in Rational Graphs* knowledge point $$ \begin{aligned} &\text{Simplified function: } g(x) = \frac{x + 9}{x + 1} \quad (x eq -3) \ &y = g(-3) = \frac{-3 + 9}{-3 + 1} = \frac{6}{-2} = -3 \end{aligned} $$ Find the discontinuities of the function. $f(x) = \frac{x^2 + 12x + 27}{x^2 + 4x + 3}$. There is a removable discontinuity at (-3, -3). { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Removable Discontinuities" ] }

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QUESTION IMAGE

Question

Explanation:

Factor the numerator and denominator

Identify the removable discontinuity

Calculate the y-coordinate of the hole

Answer: