To determine the restrictions on the domain of the composite function [tex]\( g \circ h \)[/tex], we need to analyze the functions [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex] given in the problem. The composite function [tex]\( g \circ h \)[/tex] means [tex]\( g(h(x)) \)[/tex]. Let’s go through this step-by-step:
1. Identify the given functions:
- [tex]\( g(x) = \sqrt{x - 4} \)[/tex]
- [tex]\( h(x) = 2x - 8 \)[/tex]
2. Form the composite function [tex]\( g \circ h \)[/tex]:
To form [tex]\( g \circ h \)[/tex], we substitute [tex]\( h(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(h(x)) = g(2x - 8)
\][/tex]
Now, replace [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex] with [tex]\( 2x - 8 \)[/tex]:
[tex]\[
g(2x - 8) = \sqrt{(2x - 8) - 4}
\][/tex]
Simplify inside the square root:
[tex]\[
g(2x - 8) = \sqrt{2x - 12}
\][/tex]
3. Determine the domain restrictions for [tex]\( g(2x - 8) = \sqrt{2x - 12} \)[/tex]:
The expression inside the square root must be non-negative for the square root to be defined. Hence, we set up the inequality:
[tex]\[
2x - 12 \geq 0
\][/tex]
Solve for [tex]\( x \)[/tex]:
- Add 12 to both sides:
[tex]\[
2x \geq 12
\][/tex]
- Divide both sides by 2:
[tex]\[
x \geq 6
\][/tex]
4. Conclusion:
For [tex]\( g \circ h \)[/tex] to be defined, the input [tex]\( x \)[/tex] must satisfy [tex]\( x \geq 6 \)[/tex].
Therefore, the restriction on the domain of [tex]\( g \circ h \)[/tex] is [tex]\( x \geq 6 \)[/tex].
