Chapter 2, Problem 25RE

Solution

To find the value of k and a .

To describe the portion which lies in the I and IV quadrant.

To check whether the function is even or odd or undefined for x <0

To find the rest of the curve and to graph the function.

The value of $k=4$ , $a=\frac{1}{3}$ .

Graph of the function that lies in I quadrant is $x\ge 0$ and function does not lie in IV quadrant.

The curve does not have a rest and the function is an odd function.

Given:

Function, is $f(x)=4x^{\frac{1}{3}}$ .

Concept Used:

The function $f(x)$ is said to be an even function if $f(-x)=f(x)$ and the function $f(x)$ is said to be an odd function if $f(-x)=-f(x)$ .

Calculation:

  $f(x)=4x^{\frac{1}{3}}$

Compare the function $f(x)=4x^{\frac{1}{3}}$ with $f(x)=k{x}^a$ , $k=4$ and $a=\frac{1}{3}$ .

The function $f(x)=4x^{\frac{1}{3}}\ge 0$ for all values of $x\ge 0$ and $f(x)=4x^{\frac{1}{3}}<0$ for all values of $x<0$ .

So, the function will lie in I and III quadrant only.

Now, check whether the function is even or odd or undefined for $x<0$ .

  $\begin{array}{l}f(x)=4{\left(x\right)}^{\frac{1}{3}}\\ f(-x)=4{\left(-x\right)}^{\frac{1}{3}}\\ f(-x)=-4{\left(x\right)}^{\frac{1}{3}}\\ f(-x)=-f(x)\end{array}$

Thus, the function is an odd function.

The value of the function $f(x)=4x^{\frac{1}{3}}$ vary for different values of $x$ . So, the curve does not have a rest.

The graph of the function $f(x)=4x^{\frac{1}{3}}$ is shown below:

  

From the graph it is clear that the graph lies in I and III quadrant only, function does not have a rest and the function is an odd function.

Conclusion:

The value of $k=4$ , and $a=\frac{1}{3}$ . The graph of the function that lies in I quadrant is $x\ge 0$ , the function does not have a rest and the function is an odd function.