Q. 6 - Precalculus Enhanced with Graphing Utilities

Question

True or False The cube function is odd and is increasing on the interval $(-\infty ,\infty )$. 

Step-by-Step Solution

Answer

The statement 'The cube function is odd and is increasing on the interval $(-\infty ,\infty )$.' is true.

Step 1. Given information

We have been given a statement 'The cube function is odd and is increasing on the interval $(-\infty ,\infty )$.'

We have to tell whether this statement is true or false.

Step 2. Check for the odd function

The cube function is $f\left(x\right)=x^3$.

Find $f(-x)$.

$$\begin{gathered} f(-x)={(-x)}^{3}f(-x)=-x^3 \\ f(-x)=-f\left(x\right) \end{gathered}$$

Therefore, the function is odd.

Step 3. Analyze the graph of cube function

The graph of cube function is:

It is clear from the graph that the cube function is increasing everywhere.

Therefore, the given statement is true.