Final answer:
the correct option for the domain restriction of the function g(h(x)) is not represented by the options given a) x ≥ 4, b) x ≠ 4, c) x ≤ 4, d) x > 4. The actual restriction, based on our calculation, is x ≥ 6.
Step-by-step explanation:
We are asked to find the restrictions on the domain of the composition function g(h(x)). The function g(x) is defined as sqrt(x-4), which means that the argument x-4 must be greater than or equal to zero to be in the domain of g(x). This is because the square root of a negative number is not a real number, and the domain consists of all real numbers that can be input into the function.
The function h(x) is defined as 2x-8. When we compose h(x) inside g(x), we get g(h(x)) = g(2x-8) = sqrt((2x-8) - 4) = sqrt(2x - 12). Again, for the square root to be defined, the expression 2x - 12 must be greater than or equal to zero. Hence, 2x - 12 ≥ 0, which implies x ≥ 6.
Therefore, the correct option for the domain restriction of the function g(h(x)) is not represented by the options given a) x ≥ 4, b) x ≠ 4, c) x ≤ 4, d) x > 4. The actual restriction, based on our calculation, is x ≥ 6.