Inverse Functions | MathHelp.com - By MathHelp.com
| 00:0-1 | here , were asked to graph the following function and | |
| 00:03 | use the horizontal line test to determine if it has | |
| 00:07 | an inverse , and if so , find the inverse | |
| 00:10 | function and graph it . So let's start by graphing | |
| 00:15 | the given function F of X equals two x minus | |
| 00:19 | four . And remember that F of X is the | |
| 00:22 | same as why . So we can rewrite the function | |
| 00:26 | as y equals two X -4 . Now we simply | |
| 00:31 | graph the line y equals two X -4 , Which | |
| 00:36 | has a y intercept of negative four And a slope | |
| 00:40 | of two or 2/1 . So we go up to | |
| 00:44 | and over one plot a second point and graph our | |
| 00:49 | line which will call F of X . Next we're | |
| 00:54 | asked to use the horizontal line test to determine if | |
| 00:57 | the function has an inverse , Since there's no way | |
| 01:01 | to draw a horizontal line that intersects more than one | |
| 01:05 | point on the function , the function does have an | |
| 01:09 | inverse . So we need to find the inverse and | |
| 01:13 | graph it to find the inverse . We switch the | |
| 01:17 | X and Y . In the original function Y equals | |
| 01:22 | two , X minus four . To get x equals | |
| 01:26 | two , Y -4 . Next we saw for why | |
| 01:31 | ? So we add four to both sides to get | |
| 01:34 | X plus four equals two . Why ? And divide | |
| 01:38 | both sides by two . To get one half X | |
| 01:41 | plus two equals Y . Next let's flip our equation | |
| 01:48 | so that why is on the left side and we | |
| 01:51 | have Y equals one half X plus two . Finally | |
| 01:57 | , we replace why with the notation that we use | |
| 02:01 | for the inverse function of F . As shown here | |
| 02:05 | and remember that were asked to graph the inverse as | |
| 02:08 | well . So we graph Y equals one half X | |
| 02:13 | plus two are y intercept is positive too And our | |
| 02:18 | slope is 1/2 . So we go up one and | |
| 02:22 | over two plot a second point graph the line and | |
| 02:28 | label it as the inverse function of F . Notice | |
| 02:32 | that the graph of the inverse function is a reflection | |
| 02:36 | of the original function in the line Y equals X | |
| 00:0-1 | . |
DESCRIPTION:
In this example, weâre given a relation in the form of a chart, and weâre asked to find the inverse of the relation, then graph the relation and its inverse. To find the inverse of a relation, we simply switch the x and y values in each point. In other words, the point (1, -4) becomes (-4, 1), the point (2, 0) becomes (0, 2), the point (3, 1) becomes (1, 3), and the point (6, -1) becomes (-1, 6). Next, weâre asked to graph the relation and its inverse, so letâs first graph the relation. Notice that the relation contains the points (1, -4,), (2, 0), (3, 1), and (6, -1). And the inverse of the relation contains the points (-4, 1), (0, 2), (1, 3), and (-1, 6). Finally, itâs important to understand the following relationship between the graph of a relation and its inverse. If we draw a diagonal line through the coordinate system, which is the line that has the equation y = x, notice that the relation and its inverse are mirror images of each other in this line. In other words, the inverse of a relation is the reflection of the original relation in the line y = x.
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