Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

What is the domain of [tex]f(x)=\frac{1}{x+3}[/tex]?

A. [tex](-\infty, 0) \cup (0, \infty)[/tex]
B. [tex](-\infty, 3) \cup (3, \infty)[/tex]
C. [tex](-\infty, -3) \cup (-3, \infty)[/tex]
D. [tex](-\infty, \infty)[/tex]


Sagot :

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]