To determine the domain of the function [tex]\( f(x) = \frac{x^2 - 6x + 8}{x + 2} \)[/tex], we need to identify all values of [tex]\( x \)[/tex] for which the function is defined. The function is not defined where the denominator is zero, as division by zero is undefined in mathematics.
1. Identify the denominator:
The denominator of the function is [tex]\( x + 2 \)[/tex].
2. Solve for when the denominator is zero:
Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x + 2 = 0
\][/tex]
[tex]\[
x = -2
\][/tex]
Therefore, the function [tex]\( f(x) \)[/tex] is undefined when [tex]\( x = -2 \)[/tex].
3. Determine the domain:
The domain of the function consists of all real numbers except [tex]\( x = -2 \)[/tex]. In interval notation, we express this as the union of two intervals:
- From negative infinity to -2 (excluding -2)
- From -2 to positive infinity (excluding -2)
Hence, the domain of the function in interval notation is:
[tex]\[
(-\infty, -2) \cup (-2, \infty)
\][/tex]
