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Question 1105780: Can you please help me with list two more functions that are inverses of each other? I don't understand that. And list a function that is its own inverse. I know that there are f(x) = -x, f(x) = x, f(x) = 1/x, or f(x)=-1/x but i don't if there are more besides those. Thank you so much!!!!!
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
You don't give us any idea of what you know about inverses of functions, so I'm not sure what to write in my response....
part 1: functions that are inverses of each other.
Every linear polynomial is a function that has an inverse. For example, consider the function y = 3x-2.
An algebraic way to find the inverse of a function is to switch the x and y variables and solve for the new y. For this linear function, it looks like this:
 or 
Another way to find the inverse of a given linear function is to use the fact that the inverse function "undoes" what the function does -- that is, an inverse function "gets you back where you started".
To find the inverse function for this example by this method, observe that the function does the following to the input value:
(1) multiply by 3; and
(2) subtract 2
The inverse function, to get you back where you started, has to do the opposite operations in the reverse order; for this example,
(1) add 2; and
(2) divide by 3
Applying those operations to the input value x gives you the inverse function: .
So now you have a way of finding an infinite number of pairs of functions that are inverses of each other.
Do you know that the graphs of two function that are inverses of each other are mirror images of each other with respect to the line y=x? That should make sense, since the algebraic method for finding the inverse of a function is to switch the x and y variables. In terms of the graphs of the functions, switching x and y is equivalent to reflecting the graph about the line y=x. (In reflecting a graph about the line y=x, the x axis becomes the y axis, and vice versa; so the x and y are switched.)
Here are graphs of the function (red) in the preceding example and its inverse (green), and of the line y=x (blue). You can see that the function and its inverse are mirror images of each other.
part 2: functions that are their own inverses.
You can use the fact that the graphs of a function and its inverse are mirror images of each other in the line y=x to find an infinite number of functions that are their own inverses. If a function is its own inverse, then the reflection of its graph in the line y=x must be the same line; the graph must have symmetry with respect to the line y=x.
Here are the graphs of the functions you note in your message that are their own inverses: y = -x (red), y=1/x (green), and y=-1/x (blue), along with, again the graph of y=x (purple). Note that, although you can't see it like with the other three, the graph shows that y=x is also its own inverse.
Now you get to finish this second part of your problem yourself, using the idea of a function that is its own inverse being symmetrical about the line y=x:
Do you see from the graph what other linear functions, similar to y=-x, will have symmetry with respect to the line y=x?
And what about other functions similar to y=1/x and y=-1/x that will have symmetry with respect to the line y=x?
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