Question

graphing the cube root function quick check
given $f(x) = \sqrt3{x} + 8$, identify the domain of the function as an inequality (1 point)
\\(\circ -\infty < x < \infty\\)
\\(\circ x > 0\\)
\\(\circ -10 \leq x \leq 10\\)
\\(\circ x < 0\\)
graphing calculator

Explanation:

Step1: Recall Cube Root Domain

The cube root function \( \sqrt[3]{x} \) is defined for all real numbers. This is because for any real number \( x \) (positive, negative, or zero), there is a real cube root. For example, \( \sqrt[3]{8} = 2 \), \( \sqrt[3]{-8} = -2 \), and \( \sqrt[3]{0} = 0 \).

Step2: Analyze the Given Function

The function \( f(x)=\sqrt[3]{x}+8 \) is a transformation of the basic cube root function (a vertical shift up by 8 units). Vertical shifts do not affect the domain of a function. So, the domain of \( f(x) \) is the same as the domain of \( \sqrt[3]{x} \), which is all real numbers.

Step3: Match with Options

The inequality representing all real numbers is \( -\infty < x < \infty \). The other options: \( x > 0 \) restricts to positive numbers, \( -10 \leq x \leq 10 \) restricts to a finite interval, and \( x < 0 \) restricts to negative numbers, which are all incorrect for the cube root function's domain.

Answer:

\( -\infty < x < \infty \) (the first option)