To determine the domain of the function [tex]\( f(x) = \sqrt{5x - 5} + 1 \)[/tex], we need to focus on the expression inside the square root, which is [tex]\( 5x - 5 \)[/tex]. For the square root function to yield real numbers, its argument must be non-negative (i.e., greater than or equal to zero). We need to ensure that the expression [tex]\( 5x - 5 \)[/tex] is non-negative.
Let's set up the inequality:
[tex]\[ 5x - 5 \geq 0 \][/tex]
Now, we solve this inequality for [tex]\( x \)[/tex]:
1. Add 5 to both sides of the inequality:
[tex]\[ 5x - 5 + 5 \geq 0 + 5 \][/tex]
[tex]\[ 5x \geq 5 \][/tex]
2. Divide both sides by 5:
[tex]\[ \frac{5x}{5} \geq \frac{5}{5} \][/tex]
[tex]\[ x \geq 1 \][/tex]
Therefore, the domain of the function [tex]\( f(x) = \sqrt{5x - 5} + 1 \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \geq 1 \)[/tex].
Given the options, the inequality that correctly identifies the domain of the function [tex]\( f(x) = \sqrt{5x - 5} + 1 \)[/tex] is:
[tex]\[ 5x - 5 \geq 0 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{5x - 5 \geq 0} \][/tex]
