Question

the graph shows the cube root parent function. which statement best describes the function? a. the function is increasing when x < 0. b. the function is increasing when x > 0. c. the function is always increasing. d. the function is never increasing.

Explanation:

Step1: Recall the properties of the cube - root parent function

The cube - root parent function is \(y = \sqrt[3]{x}\). Its derivative is \(y'=\frac{1}{3x^{\frac{2}{3}}}\). Since \(x^{\frac{2}{3}}=\sqrt[3]{x^{2}}>0\) for all \(x
eq0\) and \(y'\) is also positive when \(x = 0\) (by the limit definition of the derivative or by observing the graph), the slope of the function \(y=\sqrt[3]{x}\) is always non - negative.

Step2: Analyze the graph

Looking at the graph of \(y = \sqrt[3]{x}\), as \(x\) increases from \(-\infty\) to \(\infty\), the \(y\) - values also increase. For any two points \(x_1

Answer:

C. The function is always increasing.