Section 4-5:  Inverse Functions
 
 
An inverse is the operation that takes you back to where you started.  The inverse of multiplication is division, adding and subtracting, square and square root, etc.
For functions, there are two conditions for a function to be the inverse function:
   1)  g(f(x)) = x for all x in the domain of f
 
   2)  f(g(x)) = x for all x in the domain of g
 
Notice in both cases you will get back to the  element that you started with, namely, x.

The notation used to indicate an inverse function is: f -1(x) pronounced "f inverse".  This notation does not mean 1/f(x).

Example
 
1)  If f(x) = 3x - 1 and g(x) = (x + 1)/3, show that f and g are inverses to each other.
             To show that they are inverses, we must prove both of the above parts.
                     g(f(x)) = g(3x - 1) = (3x - 1 + 1)/3 = 3x/3 = x
                     f(g(x)) = f((x + 1)/3) = 3[(x + 1)/3] - 1 = x + 1 - 1 = x
Since both parts work, they are indeed inverses of each other.

To find a rule for the inverse function
 
Find the inverse function for y = 5x + 2
To find the inverse, interchange x and y.
x = 5y + 2
Now isolate for y!!
x - 2 = 5y
(x - 2)/5 = y
We now have the inverse!!
Notice, that this inverse make sense.  The original problem had adding by two and the inverse is subtracting two.  The original function had multiplying by five and the inverse has division by five.

Graphs of inverse functions
 
We have to make sure that the inverse is indeed a function.  Not all functions will have inverses that are also functions.  In order for a function to have an inverse, it must pass the horizontal line test!!
Horizontal line test
If the graph of a function y = f(x) is such that no horizontal line intersects the graph in more than one point, then f has an inverse function.
This will make sense when we discover how to graph the inverse function.  To graph the inverse function, it is simply the reflection about the line y = x.  Makes sense, because in order to get the graph, we interchange x and y.  Recall from previous sections what the reflection about the line y = x looks like.  Any two points on the same horizontal line when reflected will be on the same vertical line.  Can't have this because it wouldn't be a function.  That's why the horizontal line test works.

Example
 
1)  Find the equation of  f -1 and graph f, f -1, and y = x for f(x) = 2x - 5.
                     First, f(x) is a line and it passes the horizontal line test.
                     Find the inverse:    y = 2x - 5
                                                     x = 2y - 5
                                                     x + 5 = 2y
                                                     (x + 5)/2 = y

2)  Let f(x) = 9 - x2 for x > 0
                     Find the equation for f -1(x)
                     Sketch the graph of f, f -1, and y = x.
 
                Notice that the equation is half of a parabola.  Only the side to the right of zero.  If we tried to use the entire parabola, it wouldn't pass the horizontal line test.
                     To find the equation:  y = 9 - x2
                                                          x = 9 - y2
                                                          x - 9 = -y2
   9 - x = y2
 ,  x < 9

 
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