Final answer:
The domain of the composite function (f of g)(x) is all real numbers less than or equal to 2.5, because for f(x)=√(-x) to produce a real number, the input has to be non-positive, and the output of g(x)=2x-5 must satisfy this condition.
Step-by-step explanation:
To determine the domain of the composite function (f of g)(x), where f(x) = √(-x) and g(x) = 2x - 5, we first need to understand the domains of the individual functions f and g separately and then how they interact when composed.
For the square root function f(x), the domain is all non-positive real numbers; in other words, x must be less than or equal to zero to produce a real number result, as you cannot take the square root of a negative number in the set of real numbers. Next, we look at the function g(x), which is a linear function with an unrestricted domain of all real numbers, meaning any real number is a valid input for g. However, the output of g(x) becomes the input for f(x) when composing the functions, so we require that g(x) is less than or equal to zero.
This requirement leads to the inequality 2x - 5 ≤ 0. Solving this inequality for x gives us x ≤ ⅔ or x ≤ 2.5. Therefore, the domain of (f of g)(x) is all real numbers less than or equal to 2.5.