To determine the domain of the function [tex]\( f(x) = \sqrt{5x - 5} + 1 \)[/tex], we need to ensure that the expression inside the square root is non-negative. This is necessary because the square root of a negative number is not defined in the set of real numbers.
Let's examine the expression inside the square root: [tex]\( 5x - 5 \)[/tex]. We need to set up the inequality to ensure this expression is non-negative:
[tex]\[ 5x - 5 \geq 0 \][/tex]
Now, we solve this inequality step-by-step to find the domain:
1. Step 1: Start with the inequality:
[tex]\[ 5x - 5 \geq 0 \][/tex]
2. Step 2: Add 5 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 5x \geq 5 \][/tex]
3. Step 3: Divide both sides by 5 to solve for [tex]\( x \)[/tex]:
[tex]\[ x \geq 1 \][/tex]
The inequality we use to find the domain is:
[tex]\[ 5x - 5 \geq 0 \][/tex]
So, the domain of the function [tex]\( f(x) \)[/tex] is all values of [tex]\( x \)[/tex] that satisfy the inequality [tex]\( x \geq 1 \)[/tex].
Therefore, the domain of the function is:
[tex]\[ x \geq 1 \][/tex]
