What Values of a, b, c, and d Make f(f(x)) = x for All x?

[pyrosilver]

Homework Statement

For which numbers a, b, c, and d will the function

f(x) = (ax + b) / (cx + d)

satisfy f(f(x)) = x for all x (for which this equation makes sense)?

Homework Equations

The Attempt at a Solution

I'm not sure what to do, which is my problem. So far I think that I can plug in (ax + b) / (cx + d) for x, because that is when it will satisfy f(f(x)) = x, right?

So then I have:

http://img26.imageshack.us/img26/6364/mathnumber8.jpg

So I'm not sure where I want to end up, or where to go from here :( Can someone help me? This is due tomorrow. All help is appreciated.

 

[n!kofeyn]

Now set that equal to x (i.e., set f(f(x))=x). Then start solving for a,b,c,d.

 

[Architeuthis Dux]

To solve this problem, we need to find values for a, b, c, and d that will satisfy the equation f(f(x)) = x for all values of x. To do this, we can start by plugging in (ax + b) / (cx + d) for x in the equation f(f(x)) = x and simplifying. This will give us a new equation that we can use to solve for the values of a, b, c, and d.

Starting with f(f(x)) = x, we have:

f(f(x)) = (a((ax + b) / (cx + d)) + b) / (c((ax + b) / (cx + d)) + d) = x

Next, we can simplify this equation by multiplying the numerator and denominator by the common denominator (cx + d). This will give us:

(a(ax + b) + b(cx + d)) / (c(ax + b) + d(cx + d)) = x

Expanding the parentheses on both the numerator and denominator, we get:

(ax^2 + ab + bcx + bd) / (acx + bc + dcx + d^2) = x

Now, we can set this equation equal to x and solve for a, b, c, and d. This will involve some algebraic manipulation, but eventually we will get a solution set for a, b, c, and d that satisfies the equation f(f(x)) = x for all values of x.

I hope this helps you with your problem. If you need further assistance, please let me know. Remember to always show your work and check your solutions for validity. Good luck!