To solve the given problem, we will define the function [tex]\( g(x) \)[/tex] and evaluate it at specific points as requested.
1. Define the function [tex]\( g(x) \)[/tex]:
The function [tex]\( g(x) \)[/tex] is given by:
[tex]\[
g(x) = \frac{\sqrt{x - 3}}{2}
\][/tex]
2. Evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 3 \)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(3) = \frac{\sqrt{3 - 3}}{2} = \frac{\sqrt{0}}{2} = \frac{0}{2} = 0
\][/tex]
3. Evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 4 \)[/tex]:
Substitute [tex]\( x = 4 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(4) = \frac{\sqrt{4 - 3}}{2} = \frac{\sqrt{1}}{2} = \frac{1}{2} = 0.5
\][/tex]
4. Evaluate [tex]\( g(x - 3) \)[/tex]:
For [tex]\( g(x - 3) \)[/tex], we simply use the function [tex]\( g \)[/tex] with [tex]\( x-3 \)[/tex] substituted in place of [tex]\( x \)[/tex]:
[tex]\[
g(x - 3) = \frac{\sqrt{(x - 3) - 3}}{2} = \frac{\sqrt{x - 6}}{2}
\][/tex]
Now, let's summarize our results:
- [tex]\( g(3) = 0 \)[/tex]
- [tex]\( g(4) = 0.5 \)[/tex]
- [tex]\( g(x-3) = \frac{\sqrt{x-6}}{2} \)[/tex]
Therefore, these are the evaluated results:
1. [tex]\( g(3) = 0 \)[/tex]
2. [tex]\( g(4) = 0.5 \)[/tex]
3. [tex]\( g(x-3) \)[/tex] is the function [tex]\( \frac{\sqrt{x-6}}{2} \)[/tex]
The evaluated results are [tex]\( 0 \)[/tex], [tex]\( 0.5 \)[/tex], and [tex]\( g(x-3) = \frac{\sqrt{x-6}}{2} \)[/tex] respectively.
