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April 9th, 2013, 04:05 PM   #1
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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,
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April 9th, 2013, 06:52 PM   #2
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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?
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April 10th, 2013, 02:59 AM   #3
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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.
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April 10th, 2013, 07:09 AM   #4
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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.
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April 10th, 2013, 02:17 PM   #5
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As f(x) is its own inverse, f(f(x)) = x, so ax + ab + b = x. Hence a = 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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