To answer the question about the domain and range of the function [tex]\( g(x) = \sqrt{x - 3} \)[/tex], let's break it down step-by-step:
### Domain:
The domain of a function is the set of all possible input values (x-values) that the function can accept without causing any mathematical errors.
1. The expression inside the square root, [tex]\( x - 3 \)[/tex], must be non-negative because the square root of a negative number is not a real number.
2. Therefore, we need:
[tex]\[
x - 3 \geq 0
\][/tex]
3. Solving the inequality above:
[tex]\[
x \geq 3
\][/tex]
4. Hence, the domain is all x-values from 3 to infinity, inclusive:
[tex]\[
D: [3, \infty)
\][/tex]
### Range:
The range of a function is the set of all possible output values (y-values).
1. The function [tex]\( g(x) = \sqrt{x - 3} \)[/tex] outputs the square root of [tex]\( x - 3 \)[/tex].
2. Considering the square root function produces non-negative results, the smallest value [tex]\( g(x) \)[/tex] can attain is when [tex]\( x = 3 \)[/tex], which gives [tex]\( g(3) = \sqrt{3 - 3} = \sqrt{0} = 0 \)[/tex].
3. As [tex]\( x \)[/tex] increases beyond 3, the value inside the square root [tex]\( (x - 3) \)[/tex] becomes larger, making [tex]\( g(x) \)[/tex] larger. There is no upper bound for the values of [tex]\( g(x) \)[/tex] as [tex]\( x \)[/tex] can increase without limit.
4. Consequently, the range of [tex]\( g(x) \)[/tex] starts from 0 and goes to infinity:
[tex]\[
R: [0, \infty)
\][/tex]
### Final Answer:
The domain and range of the function [tex]\( g(x) = \sqrt{x - 3} \)[/tex] are:
[tex]\[
\text{Domain: } [3, \infty) \quad \text{and} \quad \text{Range: } [0, \infty)
\][/tex]
