Final answer:
The correct domain and range for the function 1/√x + 4 are both (0, infinity), as the function is defined for all positive real numbers and approaches zero but never reaches it.Option D is the correct answer.
Step-by-step explanation:
The correct answer is option d: The domain of the function 1/√x + 4 is (0, [infinity]) and its range is (0, [infinity]). This is because the function is defined for all positive real numbers; however, it cannot be zero since dividing by zero is undefined.
As x approaches zero, the value of the function approaches infinity, which is a vertical asymptote, while the horizontal asymptote is at y=0 since the function can come arbitrarily close to zero but never reaches it as x approaches infinity. Thus, there is no negative part in the domain or the range of the function.
The accurate choice is option d, affirming that the function (1/sqrt{x} + 4) possesses a domain of (0, ∞) and a range of (0, ∞). This is attributed to its definition for all positive real numbers, with an exclusion of zero due to the undefined nature of dividing by zero.
As x tends to zero, the function approaches infinity, establishing a vertical asymptote, while the horizontal asymptote at y=0 signifies that the function doesn't attain zero as x approaches infinity. Consequently, there exists no negative part in both the domain and range of the function.