Given f find g such that g(x)=f(x-1)

[krozer]

Homework Statement

Find two complex polynomials f and g such that f(0)=f(-1)=1, f(1)=3 and g(x)=f(x-1)

2. The attempt at a solution
Using Lagrange polynomial I got f such that f(0)=f(-1)=1, f(1)=3
Such f is defined by

f(x)=x²+x+1

Now that I've found f I need to find g such that g(x)=f(x-1), but I don't have any idea of how to do that. Any hint would be appreciated.

 

[SammyS]

Homework Statement

Find two complex polynomials f and g such that f(0)=f(-1)=1, f(1)=3 and g(x)=f(x-1)

2. The attempt at a solution
Using Lagrange polynomial I got f such that f(0)=f(-1)=1, f(1)=3
Such f is defined by

f(x)=x²+x+1

Now that I've found f I need to find g such that g(x)=f(x-1), but I don't have any idea of how to do that. Any hint would be appreciated.

Given that f(x) = x²+x+1

Find f(x-1) .

 

[HallsofIvy]

You clearly know how to do some fairly complicated Calculus so surely you know how to evaluate a function! Just replace each "x" in f(x)= x^2+ x+ 12 with x- 1.

 

[Dickfore]

Notice that if you define:
<br /> h(x) \equiv f(x) - 1<br />
then x = 0, and x = -1 are zeros of h. The simplest polynomial that can be written is:
<br /> h(x) = A x (x + 1)<br />
Then, use the fact that h(1) = f(1) - 1 = 3 - 1 = 2 to determine A. You can go back to f trivially then.

Once you have chosen a particular choice for f, finding g, as explained in the above posts, is fairly trivial (just substitute x \rightarrow x - 1, expand and simplify).