Final answer:
The composition of the functions g(x) = x² + 5 and h(x) = √x - 6 is g(h(x)) = x - 12√x + 41, and the domain of this composite function is [6, ∞) in interval notation.
Step-by-step explanation:
The task is to find the composition of functions g(x) = x² + 5 and h(x) = √x - 6. When we compose two functions, we substitute one function into the other. In this case, to find g(h(x)), also denoted as g ∙ h, we substitute h(x) into g(x). So, g(h(x)) = g(√x - 6) = (√x - 6)² + 5. This simplifies to g(h(x)) = x - 12√x + 41 since (√x - 6)^2 = x - 12√x + 36 and adding 5 gives us 41. The domain of the composition g(h(x)) depends on the domain of h(x) because it must be defined first before applying g(x). The function h(x) is defined for x ≥ 6 because we cannot take the square root of a negative number. Therefore, the domain of g ∙ h is [6, ∞) in interval notation.