Chapter 2, Problem 28RE

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 $k=\frac{2}{3}$ , and $a=-4$ . The graph of the function lies in II and IV quadrants only, the function is an even function and the function does not have a rest

Given:

Function, is $f(x)=\frac{2}{3}{x}^{-4}$ .

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:

Compare the function $f(x)=\frac{2}{3}{x}^{-4}$ with $f(x)=k{x}^a$ , $k=\frac{2}{3}$ and $a=-4$ .

The function $f(x)=\frac{2}{3}{x}^{-4}>0$ for all values of $x>0$ and $x<0$

The function,

  $\begin{array}{c}f(x)=\frac{2}{3}{x}^{-4}\\ =\frac{2}{3x^4}\end{array}$ is not defined for $x=0$

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

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

  $\begin{array}{c}f(x)=\frac{2}{3}{\left(x\right)}^{-4}\\ f(-x)=\frac{2}{3}{\left(-x\right)}^{-4}\\ f(-x)=\frac{2}{3}{\left(x\right)}^{-4}\\ f(-x)=f(x)\end{array}$

Thus, the function is an even function.

The value of the function $f(x)=\frac{2}{3}{x}^{-4}$ vary for different values of $x$ . But function value approaches to infinity for large values of $x$ So, the curve does not have a rest

The graph of the function $f(x)=\frac{2}{3}{x}^{-4}$ is as shown below:

  

From the graph it is clear that the graph lies in II and IV quadrants only, function does not have a rest and the function is an even function.

Conclusion:

The value of $k=\frac{2}{3}$ , and $a=-4$ . The graph of the function lies in II and IV quadrants only, the function does not have a rest. and the function is an even function.