Question

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

Short answer

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

Step-by-step solution

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.