Final answer:
Composition of the given functions results in (f \circ g)(x) = -30x - 75, and (g \circ f)(x) = -30x + 54. Both compositions have a domain of all real numbers (ℝ). Thus, options a and b provided in the question are incorrect.
Step-by-step explanation:
To find the composition of functions (f \circ g)(x) and (g \circ f)(x), we simply substitute the inside function into the outside function and simplify. For the functions given, f(x) = -10x + 15 and g(x) = 3x + 9, let's perform the substitutions.
(f \circ g)(x) means plugging g(x) into f(x). So:
f(g(x)) = f(3x + 9) = -10(3x + 9) + 15 = -30x - 90 + 15 = -30x - 75.
Therefore, option a is incorrect.
For (g \circ f)(x), we plug f(x) into g(x):
g(f(x)) = g(-10x + 15) = 3(-10x + 15) + 9 = -30x + 45 + 9 = -30x + 54.
Therefore, option b is incorrect.
The domain for both composite functions is \(\mathbb{R}\), the set of all real numbers, because there are no restrictions such as division by zero or square roots of negative numbers in either composed function.