How Do You Solve This Composite Function Problem?

[songoku]

Homework Statement

Find f(x) given that f ( f(x) - x²) = x² - 5x + 3

Homework Equations

Not sure

The Attempt at a Solution

I tried assuming f(x) = ax + b and use composite function but end up wrong. Please give me idea to start

Thanks

 

[SammyS]

Homework Statement

Find f(x) given that f ( f(x) - x²) = x² - 5x + 3

Homework Equations

Not sure

The Attempt at a Solution

I tried assuming f(x) = ax + b and use composite function but end up wrong. Please give me idea to start

Thanks

It seems reasonable to assume that f(x) is a polynomial.

If you compose linear function and a quadratic you get a quadratic function.

One way to do that is for f(x) to be linear, which is what you tried & you say doesn't work.

Another way is for f(x) to be a quadratic such that f(x) - x² is linear.

Try that.

 

[songoku]

It seems reasonable to assume that f(x) is a polynomial.

If you compose linear function and a quadratic you get a quadratic function.

One way to do that is for f(x) to be linear, which is what you tried & you say doesn't work.

Another way is for f(x) to be a quadratic such that f(x) - x² is linear.

Try that.

Thank you very much

 

[haruspex]

To turn Sammy's remarks into a general approach, suppose f is a polynomial degree n. What are the possible degrees of f(x)-x²? What, then, are the possible degrees of f(f(x)-x²)?