Question

cubic & cube root functions cubic & cube root functions unit test back to intro page
$f(x) = \sqrt3{x}$
-10 -5 0 5 10
-5 0 5
which of the following does not represent the domain of the function $g(x) = \sqrt3{x}$?
option #1: $(-\infty, \infty)$
option #2: $-\infty < x < \infty$
option #3: $-\infty, \infty$
(1 point)
the best answer is option #

Explanation:

Step1: Recall domain of cube root function

The function \( g(x)=\sqrt[3]{x} \) has a domain of all real numbers, since we can take the cube root of any real number (positive, negative, or zero). The domain in interval notation can be written as \( (-\infty, \infty) \) or in inequality notation as \( -\infty < x < \infty \).

Step2: Analyze each option

• Option #1: \( (-\infty, \infty) \) is correct interval notation for all real numbers.
• Option #2: \( -\infty < x < \infty \) is correct inequality notation for all real numbers.
• Option #3: \( [-\infty, \infty] \) is incorrect. Interval notation uses parentheses for \( \infty \) (and \( -\infty \)) because they are not actual numbers that can be included (we can't "reach" infinity), so we use parentheses, not brackets.

Answer:

Option #3: \([-\infty, \infty]\)