Sure, let's find the domain of the function [tex]\( f(x) = \frac{x+6}{6-x} \)[/tex].
1. Identify the Denominator and its Restrictions: To determine the domain of the function, we need to consider when the function is undefined. A function is typically undefined where its denominator is zero (since division by zero is undefined). Thus, we first identify the denominator of the function:
[tex]\[
\text{Denominator: } 6 - x
\][/tex]
2. Set the Denominator to Zero: Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
6 - x = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]: Solving the equation for [tex]\( x \)[/tex]:
[tex]\[
6 - x = 0 \implies x = 6
\][/tex]
This indicates that at [tex]\( x = 6 \)[/tex], the denominator becomes zero, making the function undefined at this value.
4. Determine the Domain: The function is valid for all real numbers except where it is undefined. Hence, the domain of the function excludes [tex]\( x = 6 \)[/tex].
5. Express the Domain: So, the domain of the function [tex]\( f(x) = \frac{x+6}{6-x} \)[/tex] is all real numbers except [tex]\( x = 6 \)[/tex]. We can write this in interval notation as:
[tex]\[
\{ x \in \mathbb{R} \mid x \neq 6 \}
\][/tex]
Therefore, the domain of the function is:
[tex]\[
x \in \mathbb{R}, x \neq 6
\][/tex]
In interval notation, we can express the domain as:
[tex]\[
(-\infty, 6) \cup (6, \infty)
\][/tex]
