Chapter 1.6, Problem 64E

To determine

To state: that whether the function is one-to-one or not and verify graphically.

Answer to Problem 64E

No, the function is not one-to-one.

Explanation of Solution

Given information : thefunction is,

  $h\left(x\right)=\frac{4}{x^2}$

let $h\left(x\right)=h\left(y\right)$ ,

  $\begin{array}{l}h\left(x\right)=h\left(y\right)\\ \frac{4}{x^2}=\frac{4}{y^2}\text{ [substitute }h\left(x\right)=\frac{1}{x^2}]\\ y^2=x^2\text{ [cross multiplication]}\\ x\ne y\end{array}$

There is many possibilities that $x\text{ and }y$ , are not equal. So the value of y has occur more than one time for x -value. So the function is not one-to-one.

to graph the given function first create a table for plotting the points and then make a curve smoothly by jointing the points.

x	y
1	4
-1	4
2	1
-2	1

The graph can be obtained as:

  

From the above graph, it can be observed that the graph of the function hashorizontal lines that intercept the graph more than one times, so the function isnot a one-to-one and its inverse functiondoes not exists.