Okay, I’m writing this because a lot of people in my math class are having problems with this and because I still have (some) faith in humanity I’ll assume there difficulty is due to a lack of sources from which to learn. So today’s topic is “Inverse Functions”…
So to start a inverse function “inverts” the input/output of a function. For instance if a function does this:
[math]f(x) doubleright b[/math]
Then the inverse would do this:
[math]{f^{-1}}(b) doubleright x[/math]
So basically the input into the first function is the output of the second function. Also note that the inverse function (the second one) has a [math]{f^{-1}}(x)[/math] instead of [math]f(x)[/math] This is how you donate that the function is the inverse. That said, lets look at some examples. The most basic of which would look like so.
[math]f(x) = x^2[/math]
[math]{f^{-1}}(x)= sqrt{x}[/math]
That should be very easy to follow. basically the first function takes a number and squares it the second takes the squared number and turns it into the starting number. In order to find the inverse of a function one needs to simply swap the “f(x)” and the “x” and solve for “x” again. Here’s a quick example starting and the first function and following all of the steps to get to the inverse.

[math]f(x)={(x-2)^2}/{21}[/math]
[math]x={(f(x)-2)^2}/{21}[/math]
[math]21x={(f(x)-2)^2}[/math]
[math]sqrt{21x}=f(x)-2[/math]
[math]{f^{-1}}(x)=(sqrt{21x})+2[/math]

So then, using 5 as our “x” value the starting function gives us .4285714286 putting that answer into the inverse function gives us 5. So the inverse simply takes the output of the first function and tells you that the input into the first function was (please note that the provided example only works with positive numbers). Hopefully this will help anyone confused by this subject.