Question

what is the domain of the function $f(x) = \frac{x + 1}{x^2 - 6x + 8}$?
all real numbers except $-4$ and $-2$
all real numbers except $2$ and $4$
all real numbers
all real numbers except $-1$

Explanation:

Step1: Find denominator's roots

The function is a rational function, so the denominator cannot be zero. Factor the denominator: \(x^2 - 6x + 8=(x - 2)(x - 4)\).
Set denominator equal to zero: \((x - 2)(x - 4)=0\).

Step2: Solve for x

Using zero - product property, if \(ab = 0\), then \(a = 0\) or \(b = 0\). So \(x-2 = 0\) gives \(x = 2\) and \(x - 4=0\) gives \(x = 4\).
The domain of a rational function is all real numbers except the values that make the denominator zero. So the domain is all real numbers except \(x = 2\) and \(x = 4\).

Answer:

all real numbers except 2 and 4