To solve the problem, let's break it down step by step.
First, we need to understand the composition of the two functions [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex].
Given:
[tex]\[ g(x) = \sqrt{x - 4} \][/tex]
[tex]\[ h(x) = 2x - 8 \][/tex]
We want to find [tex]\( g(h(10)) \)[/tex].
Step 1: Calculate [tex]\( h(10) \)[/tex]
Substitute [tex]\( x = 10 \)[/tex] into the function [tex]\( h(x) \)[/tex]:
[tex]\[ h(10) = 2(10) - 8 \][/tex]
[tex]\[ h(10) = 20 - 8 \][/tex]
[tex]\[ h(10) = 12 \][/tex]
Step 2: Calculate [tex]\( g(h(10)) \)[/tex]
Now substitute [tex]\( h(10) = 12 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(12) = \sqrt{12 - 4} \][/tex]
[tex]\[ g(12) = \sqrt{8} \][/tex]
Simplifying [tex]\( \sqrt{8} \)[/tex]:
[tex]\[ \sqrt{8} = \sqrt{4 \cdot 2} \][/tex]
[tex]\[ \sqrt{8} = \sqrt{4} \cdot \sqrt{2} \][/tex]
[tex]\[ \sqrt{8} = 2\sqrt{2} \][/tex]
Therefore,
[tex]\[ g(h(10)) = 2\sqrt{2} \][/tex]
Given the options:
- [tex]\( 2\sqrt{2} \)[/tex]
- [tex]\( \sqrt{6} \)[/tex]
- [tex]\( \sqrt{6} - 8 \)[/tex]
- [tex]\( 2\sqrt{6} - 8 \)[/tex]
The correct answer is:
[tex]\[ 2\sqrt{2} \][/tex]
