To determine the domain of the function [tex]\(\sqrt[4]{3 t - 3}\)[/tex], we must ensure that the expression inside the fourth root is non-negative because the fourth root of a negative number is not real.
The steps to find the domain are as follows:
1. Set up the inequality: To find the domain, the expression inside the fourth root, [tex]\(3t - 3\)[/tex], must be greater than or equal to zero:
[tex]\[
3t - 3 \geq 0
\][/tex]
2. Solve the inequality: Solve the inequality for [tex]\(t\)[/tex]:
[tex]\[
3t - 3 \geq 0
\][/tex]
Add 3 to both sides of the inequality:
[tex]\[
3t \geq 3
\][/tex]
Divide both sides by 3:
[tex]\[
t \geq 1
\][/tex]
3. Write the domain: The solution to the inequality is [tex]\(t \geq 1\)[/tex]. This means that the value of [tex]\(t\)[/tex] must be at least 1. In interval notation, the domain of the function is:
[tex]\[
[1, \infty)
\][/tex]
Therefore, the domain of the function [tex]\(\sqrt[4]{3 t - 3}\)[/tex] is [tex]\( t \in [1, \infty) \)[/tex].
