To determine the domain of the function [tex]\( f(x) = \frac{5x + 3}{x} \)[/tex], we need to identify the values of [tex]\( x \)[/tex] for which the function is defined.
1. Identify the Restrictions:
- The denominator of the function is [tex]\( x \)[/tex].
- A function is undefined whenever the denominator is zero because division by zero is not defined in mathematics.
2. Set the Denominator Not Equal to Zero:
- For the function [tex]\( f(x) \)[/tex], we need to exclude any [tex]\( x \)[/tex] that makes the denominator zero.
- Therefore, we set the denominator [tex]\( x \neq 0 \)[/tex].
3. Describe the Domain:
- Since [tex]\( x \)[/tex] cannot be zero, the function is defined for all other real numbers.
4. Write the Domain in Interval Notation:
- We exclude the point where [tex]\( x = 0 \)[/tex].
- Therefore, the domain is all real numbers except [tex]\( x = 0 \)[/tex].
In interval notation, this is expressed as:
[tex]\[
(-\infty, 0) \cup (0, \infty)
\][/tex]
