Consider the function [tex][tex]$f(x)=\sqrt{5x-5}+1$[/tex][/tex].

1. Which inequality is used to find the domain?
A. [tex][tex]$5x-4 \geq 0$[/tex][/tex]
B. [tex][tex]$\sqrt{5x-5}+1 \geq 0$[/tex][/tex]
C. [tex][tex]$5x \geq 0$[/tex][/tex]
D. [tex][tex]$5x-5 \geq 0$[/tex][/tex]

2. What is the domain of the function?
Domain: [tex]x \geq 1[/tex]



Answer :

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]