Question

question 5 consider the rational function $g(x)=\frac{3x + 12}{x^{2}+7x + 12}$ factor the numerator to get factor the denominator to get the rational function $f$ has a vertical asymptote at $x=$ question help: message instructor submit question

Explanation:

Step1: Factor the numerator

Factor out the greatest - common factor from \(3x + 12\). The GCF of \(3x\) and \(12\) is \(3\), so \(3x+12 = 3(x + 4)\).

Step2: Factor the denominator

We need to find two numbers that multiply to \(12\) and add up to \(7\). The numbers are \(3\) and \(4\). So \(x^{2}+7x + 12=(x + 3)(x+4)\).

Step3: Find the vertical asymptote

The vertical asymptotes of a rational function \(y=\frac{f(x)}{g(x)}\) occur at the values of \(x\) that make the denominator equal to zero, while the numerator is non - zero. Set the denominator \((x + 3)(x + 4)=0\). Solving \(x+3 = 0\) gives \(x=-3\) and solving \(x + 4=0\) gives \(x=-4\). But when \(x=-4\), the numerator \(3(x + 4)=0\) too. So the vertical asymptote is \(x=-3\).

Answer:

Factor the numerator to get \(3(x + 4)\)
Factor the denominator to get \((x + 3)(x + 4)\)
The rational function \(f\) has a vertical asymptote at \(x=-3\)