To determine the domain of the function [tex]\( f(x) = \frac{1}{x+3} \)[/tex], we need to identify all the values of [tex]\(x\)[/tex] for which the function is defined.
1. The function [tex]\( f(x) = \frac{1}{x+3} \)[/tex] is a rational function. A rational function is undefined wherever its denominator is zero.
2. To find the values of [tex]\(x\)[/tex] that make the denominator zero, we solve the equation:
[tex]\[
x + 3 = 0
\][/tex]
3. Solving for [tex]\(x\)[/tex]:
[tex]\[
x = -3
\][/tex]
4. Thus, the function [tex]\( f(x) = \frac{1}{x+3} \)[/tex] is undefined when [tex]\( x = -3 \)[/tex].
5. Therefore, the domain of the function consists of all real numbers except [tex]\( x = -3 \)[/tex]. This can be written in interval notation as:
[tex]\[
(-\infty, -3) \cup (-3, \infty)
\][/tex]
Thus, the correct answer is:
C. [tex]\( (-\infty, -3) \cup (-3, \infty) \)[/tex]
