Question

how many vertical asymptotes does the function have?
\\( \frac { ( x + 3 ) ( x + 2 ) } { ( x - 2 ) ( x + 1 ) } \\)

Explanation:

Step1: Find the denominator's roots

Set \((x - 2)(x + 1)=0\).
Using the zero - product property \(ab = 0\Rightarrow a=0\) or \(b = 0\), we get \(x-2=0\Rightarrow x = 2\) and \(x + 1=0\Rightarrow x=-1\).

Step2: Check for common factors

The numerator is \((x + 3)(x + 2)\). There are no common factors between the numerator \((x + 3)(x + 2)\) and the denominator \((x - 2)(x + 1)\).

Step3: Count vertical asymptotes

The vertical asymptotes of a rational function \(y=\frac{f(x)}{g(x)}\) (where \(f(x)\) and \(g(x)\) are polynomials) occur at the values of \(x\) that make \(g(x)=0\) (when there are no common factors). Since the denominator \((x - 2)(x + 1)\) has two distinct roots \(x = 2\) and \(x=-1\) with no cancellation with the numerator, the number of vertical asymptotes is 2.

Answer:

2