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April 9th, 2013, 04:05 PM  #1 
Newbie Joined: Apr 2013 Posts: 7 Thanks: 0  Find all linear function given a function equals its inverse
Hello, I'm not sure exactly what I need to do for the following question  Find all linear functions f(x) = ax + b such that f(x) = f^1 (x), where f(x) = (x  3)^2(4  x)(2x + 1) Anyone able to give me a clue where to start? Thank you, 
April 9th, 2013, 06:52 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,901 Thanks: 1095 Math Focus: Elementary mathematics and beyond  Re: Find all linear function given a function equals its inv
You have conflicting definitions for f(x) and (x  3)^2(4  x)(2x + 1) is not invertible over the domain of real numbers. Is this exactly how the problem appears?

April 10th, 2013, 02:59 AM  #3 
Newbie Joined: Apr 2013 Posts: 7 Thanks: 0  Re: Find all linear function given a function equals its inv
'Find all linear functions f(x) = ax + b such that f(x) = f^1(x)" is how the question appears. (x3)^2(4x)(2x+1) is defined in a previous question and I thought it carried over into this one, but I'm probably wrong. I had assumed F(x) = ax + b was just the form of the linear function I was to attempt to find.

April 10th, 2013, 07:09 AM  #4 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Find all linear function given a function equals its inv
Yes, any linear equation can be written as f(x)= y= ax+ b. To find such functions that satisfy , you need, certainly, first to find the inverse function! You can do that by "swapping" y and x: solve x= ay+ b for y, then set the two equal. Being equal, for functions, means equal for all x so choosing two different values of x will give you two equations to solve for a and b. Of course, the problem says "find all linear functions" so you cannot find specific values for a and b.

April 10th, 2013, 02:17 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,116 Thanks: 1910 
As f(x) is its own inverse, f(f(x)) = x, so a²x + ab + b = x. Hence a² = 1. If a = 1, b must be 0 and f(x) = x. If a = 1, b can have any value and f(x) = x + b. 

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equals, find, function, inverse, linear 
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