Exercise 6 Page 645

We are asked to complete the given table.

Equation	Is the inverse of a function?		Is the function its own inverse?
	Yes	No	Yes	No
$y=\text{-}x$
$y=3\ln x+2$
$y={\left(\frac{1}{x}\right)}^2$
$y=\frac{x}{x-1}$

Let's start with the first equation $y=\text{-}x.$

$y=\text{-}x$

We will try to find the inverse of $y=\text{-}x.$ First, let's switch $x$ with $y.$ Now, try to solve for $y.$ We found that the equation $y=\text{-}x$ is the inverse of a function $y=\text{-}x.$ This means that the function is its own inverse! Let's mark that in the given table.

Equation	Is the inverse of a function?		Is the function its own inverse?
	Yes	No	Yes	No
$y=\text{-}x$	$✓$		$✓$
$y=3\ln x+2$
$y={\left(\frac{1}{x}\right)}^2$
$y=\frac{x}{x-1}$

$y=3\ln x+2$

Next we will try to find the inverse of $y=3\ln x+2.$ Once again, we can start by switching $x$ with $y.$ Let's solve this for $y$! Now, write this in exponential form. We found that the inverse function of $y=3\ln x+2$ is $y=e^{\frac{x-2}{3}}.$ These functions are not the same, and therefore $y=3\ln x+2$ is not its own inverse. Let's mark that in our table.

Equation	Is the inverse of a function?		Is the function its own inverse?
	Yes	No	Yes	No
$y=\text{-}x$	$✓$		$✓$
$y=3\ln x+2$	$✓$			$\times$
$y={\left(\frac{1}{x}\right)}^2$
$y=\frac{x}{x-1}$

$y={\left(\frac{1}{x}\right)}^2$

This time we will try to find the inverse of $y=\left(\frac{1}{x}\right).$ Similarly to before, we will first switch $x$ with $y.$ Let's try to solve this for $y.$ We found the inverse of the equation, but something is not right. If this were a function, then each input $x$ would be assigned to exactly one output $y.$ This is not the case here — $x$ is assigned to two outputs, $\frac{1}{\sqrt{x}}$ and $\text{-}\frac{1}{\sqrt{x}}.$ This means that the inverse equation might not be a function. Let's use the Horizontal Line Test to find if the inverse of $y={\left(\frac{1}{x}\right)}^2$ is a function. To use it, we will first need to graph the equation.

Now, let's check if a horizontal line can intersect the graph more than once.

As we can see, there is a line that intersects the graph more than once. Therefore, the equation has no inverse function, which also means that the equation is definitely not the inverse of itself. Let's write that in our table.

Equation	Is the inverse of a function?		Is the function its own inverse?
	Yes	No	Yes	No
$y=\text{-}x$	$✓$		$✓$
$y=3\ln x+2$	$✓$			$\times$
$y={\left(\frac{1}{x}\right)}^2$		$\times$		$\times$
$y=\frac{x}{x-1}$

$y=\frac{x}{x-1}$

Finally, we will check whether the inverse of $y=\frac{x}{x-1}$ is a function. One last time, we will switch $x$ with $y.$ Let's try to solve this for $y.$ We found that the inverse of $y=\drac{x}{x-1}$ is $y=\frac{x}{x-1}.$ We can complete our table.

Equation	Is the inverse of a function?		Is the function its own inverse?
	Yes	No	Yes	No
$y=\text{-}x$	$✓$		$✓$
$y=3\ln x+2$	$✓$			$\times$
$y={\left(\frac{1}{x}\right)}^2$		$\times$		$\times$
$y=\frac{x}{x-1}$	$✓$		$✓$