How Do You Solve for a Linear Function Where f(f(x)) Equals 4x + 3?

[Gray Matter]

Homework Statement

The problem is this: "Find a linear function f such that (f o f)(x) = 4x + 3.

Homework Equations

The only hint I've received is "write f(x) = ax + b, and use the given equation to find a and b."

The Attempt at a Solution

I've been staring at this thing for almost half an hour now. I'm most confused because I need to find an equation that when composed with itself equals 4x + 3.

I've tried to solve for a and b by substituing the 4 and 3 from the final equation as the various variables (a, x, and b), but all those solutions have left me with nothing useful. I also don't understand how I am supposed to solve for a and b when I only have one equation.

I obviously don't want the entire solution printed for me, but believe me, I have tried and I cannot get past this problem. Even just a little tip would be great, thanks!

 

[HallsofIvy]

Well, stop staring and start doing! That's the key to mathematics- just staring at a problem does not cause the answer to suddenly pop into your head (and certainly not on your paper!).

You say you tried "substituting the 4 and 3 from the final equation as the various variables (a, x, and b) but there is no point in doing that. There is no reason to think that a and b will be the same as the coefficients for the final equation and they certainly won't be "x". Plugging numbers in at random is no better a method of solving a math problem than staring at the paper!

What is a good method is thinking about the definition of the thing you are working with. If f(x)= ax+ b, what is f o f(x)? I assume you know that f o f(x)= f(f(x)). If f(x)= ax+ b, then f(f(x))= f(ax+ b)= a(ax+ b)+ b. Now set that equal to 4x+ 3 and solve for a and b.