Chapter 1.6, Problem 12E

To determine

Whether the function $g\left(x\right)=\sqrt[3]{x}$ is one-to-one.

Answer to Problem 12E

The function $g\left(x\right)=\sqrt[3]{x}$ is one-to one.

Explanation of Solution

Express the given function as $g\left(x\right)=x^{\frac{1}{3}}$.

Use horizontal line test to determine whether the function $g\left(x\right)=x^{\frac{1}{3}}$ is one-to-one.

Horizontal line test states that the graph of the function is one-to-one function if and only if a horizontal line intersects the graph exactly once.

Sketch the graph of the function $g\left(x\right)=x^{\frac{1}{3}}$ and perform the horizontal line test as shown below in Figure 1.

From Figure 1, it is observed that the horizontal line intersects the curve exactly at once which means it passes the horizontal line test. Therefore, the function $g\left(x\right)=\sqrt[3]{x}$ is one-to-one.