Find the domain and range of the inverse of the given function.

f(x) =

Question

Find the domain and range of the inverse of the given function.

f(x) =

BlackTom AI · Accepted Answer

To determine the domain and range of the inverse, first recall the original function f(x) = sqrt(x - 4).
Since the square root requires its input to be nonnegative, the domain of f is x ≥ 4. The output of f is a nonnegative number, so the range of f is y ≥ 0.
When forming the inverse, we swap domain and range: the domain of f^{-1} becomes the range of f, and the range of f^{-1} becomes the domain of f.
Thus, f^{-1} has domain [0, ∞) and range [4, ∞).
Now evaluate each option:

Option 1: Domain: [4, ∞); range: [0, ∞).
This option swaps the actual domain and range of the inverse, which is incorrect because the inverse’s domain should be the original range [0, ∞), not [4, ∞).

Option 2: Domain and range: all real numbers.
This is incorrect because the inverse cannot take negative inputs (the original range was y ≥ 0), so the domain cannot be all real numbers, and the range cannot be all real numbers either since the original domain was restricted to x ≥ 4.

Option 3: Domain: [4, ∞); range: [4, ∞).
This option incorrectly uses [4, ∞) for the domain instead of [0, ∞). The range [4, ∞) is correct for the inverse, but the domain is not.

Option 4: Domain: [0, ∞); range: [4, ∞).
This matches the swapped roles of domain and range from the original function: the inverse can accept nonnegative inputs and outputs values starting at 4 and up. This is the consistent and correct description of f^{-1}.

Find the domain and range of the inverse of the given function. f(x) = 单项选择题

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