Describing Inverses of Functions

We can use the Horizontal Line Test to determine whether the inverse of a function is also a function. If it is a function, $(f (x)$ is invertible. To do this with the given function, we will first make a table of values to graph it. When making a table of values, make sure to use a variety of points, including negative and positive values.

$x$	$2 x^{2}$	$f (x) = 2 x^{2}$
$- 4$	$2 (- 4)^{2}$	$32$
$- 3$	$2 (- 3)^{2}$	$18$
$- 2$	$2 (- 2)^{2}$	$8$
$- 1$	$2 (- 1)^{2}$	$2$
$0$	$2 (0)^{2}$	$0$
$1$	$2 (1)^{2}$	$2$
$2$	$2 (2)^{2}$	$8$
$3$	$2 (3)^{2}$	$18$

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 positive leading coefficient. This tells us about the end behavior of the function. $f (x) \to - \infty as x \to + \infty f (x) \to + \infty as x \to + \infty$ Therefore, it will continue upward in both directions.

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. The function $f (x)$ is not invertible.

Describing Inverses of Functions 1.8 - Solution