To find the domain of the rational function [tex]\( s(t) = \frac{t^2 + 5}{2t} \)[/tex], follow these steps:
1. Identify the Denominator: The denominator of the rational function is [tex]\( 2t \)[/tex].
2. Determine When the Denominator is Zero:
- Set the denominator equal to zero and solve for [tex]\( t \)[/tex]:
[tex]\[
2t = 0
\][/tex]
- Solving for [tex]\( t \)[/tex], we get:
[tex]\[
t = 0
\][/tex]
3. Exclude the Values that Make the Denominator Zero: Since the denominator [tex]\( 2t \)[/tex] equals zero when [tex]\( t = 0 \)[/tex], [tex]\( t = 0 \)[/tex] is not included in the domain. If [tex]\( t = 0 \)[/tex], the function becomes undefined because division by zero is not possible.
4. State the Domain: The function [tex]\( s(t) = \frac{t^2 + 5}{2t} \)[/tex] is defined for all real numbers except where the denominator is zero. Therefore, the domain of [tex]\( s(t) \)[/tex] includes all real numbers except [tex]\( t = 0 \)[/tex].
Hence, the domain of the rational function [tex]\( s(t) = \frac{t^2 + 5}{2t} \)[/tex] is:
[tex]\[
\text{All real numbers except } t = 0.
\][/tex]
In interval notation, this can be written as:
[tex]\[
(-\infty, 0) \cup (0, \infty)
\][/tex]
So, the domain of [tex]\( s(t) = \frac{t^2 + 5}{2t} \)[/tex] is all real numbers except [tex]\( t = 0 \)[/tex].
