To find the restrictions on the domain of the composition of functions \( g \circ h \), we need to understand the conditions under which the composition \( g(h(x)) \) is defined. Here is a step-by-step solution:
1. Identify the functions:
- The function \( g \) is given by \( g(x) = \sqrt{x - 4} \).
- The function \( h \) is given by \( h(x) = 2x - 8 \).
2. Express the composition \( g \circ h \):
- To form the composition \( g(h(x)) \), substitute \( h(x) \) into \( g(x) \).
- This gives us \( g(h(x)) = g(2x - 8) \).
3. Substitute and simplify:
- Substitute \( h(x) \) into \( g(x) \): \( g(2x - 8) = \sqrt{(2x - 8) - 4} \).
- Simplify the expression inside the square root: \( g(2x - 8) = \sqrt{2x - 12} \).
4. Determine the domain restrictions:
- The expression under the square root, \( 2x - 12 \), must be non-negative for the square root to be defined.
- This means \( 2x - 12 \geq 0 \).
5. Solve the inequality:
- Add 12 to both sides: \( 2x \geq 12 \).
- Divide both sides by 2: \( x \geq 6 \).
Therefore, the restriction on the domain of \( g \circ h \) is:
[tex]\[ x \geq 6 \][/tex]
