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 April 9th, 2013, 03:05 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, 05:52 PM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,932 Thanks: 1127 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, 01:59 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. (x-3)^2(4-x)(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, 06:09 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 $f^{-1}(x)= f(x)$, 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, 01:17 PM #5 Global Moderator   Joined: Dec 2006 Posts: 20,474 Thanks: 2039 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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