Exercise 45 Page 397

We can use the Horizontal Line Test to determine whether the inverse of a function is also a function. To do this with the given function, we will first make a table of values to graph it. Make sure to use a variety of points, including negative and positive values.

$x$	$\text{-}2x^4-x-2$	$h(x)=\text{-}2x^4-x-2$
$\text{-}3$	$\text{-}2(\text{-}3{)}^4-(\text{-}3)-2$	$\text{-}161$
$\text{-}2$	$\text{-}2(\text{-}2{)}^4-(\text{-}2)-2$	$\text{-}32$
$\text{-}1$	$\text{-}2(\text{-}1{)}^4-(\text{-}1)-2$	$\text{-}3$
$0$	$\text{-}2(0{)}^4-0-2$	$\text{-}2$
$1$	$\text{-}2(1{)}^4-1-2$	$\text{-}5$
$2$	$\text{-}2(2{)}^4-2-2$	$\text{-}36$
$3$	$\text{-}2(3{)}^4-3-2$	$\text{-}167$

Now we can plot the obtained points and connect them with a smooth curve. Consider also that this is an even-degree polynomial with a negative leading coefficient. This tells us about the end behavior of the function. Let's draw the function.

Finally, we can perform the Horizontal Line Test. If the horizontal lines intersect the graph once, then the inverse is also a function. Conversely, if there is even one horizontal line that intersects the graph more than once, then the inverse is not a function.

We can see above that there are horizontal lines that intersect the curve at more than one point. Therefore, the inverse of the given function is not a function.