Q. 7 - Precalculus Enhanced with Graphing Utilities

Question

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

Step-by-Step Solution

Answer

The statement 'The cube root function is odd and is decreasing on the interval $(-\infty ,\infty )$.' is false.

Step 1. Given information

We have been given a statement 'The cube root function is odd and is decreasing 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 $f\left(x\right)=\sqrt[3]{x}$.

Find $f(-x)$.

$f(-x)=\sqrt[3]{-x}$

Since, $f(-x) \mathrm{is} \mathrm{not} \mathrm{equal} \mathrm{to} -f\left(x\right).$

This is not an odd function.

Step 3. Analyze the graph of cube root function

The graph of cube root function is:

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

Therefore, the given statement is false.