Question

consider the function $g(x) = \sqrt3{x - 4}$
what is the range of this functions inverse?
$\bigcirc\\ (-infty, 0)$
$\bigcirc\\ (-infty, infty)$
$\bigcirc\\ (0, infty)$
$\bigcirc\\ (-infty, -4)$

Explanation:

Step1: Recall the relationship between a function and its inverse

The range of a function's inverse is equal to the domain of the original function.

Step2: Determine the domain of the original function \( g(x) = \sqrt[3]{x - 4} \)

For cube root functions, the expression inside the cube root (the radicand) can be any real number because the cube root of a negative number is defined (unlike the square root of a negative number, which is not a real number). So, the domain of \( g(x) \) is all real numbers, which is \( (-\infty, \infty) \).

Step3: Conclude the range of the inverse function

Since the range of the inverse function is the domain of the original function, the range of \( g^{-1}(x) \) is \( (-\infty, \infty) \).

Answer:

\((-\infty, \infty)\) (corresponding to the option "(-∞, ∞)")