Hello Pawan --
I'm having some difficulty understanding what is meant transformation. I've been given the equations :
f(x)= x^3-2x^2+x and g(x)= 3f(x+1)-2
Now, I've received your explanation of how f(x+1) = x^3 + 3x^2 + 3x +1. This is fine, BUT what I don't understand is how to LIST the transformation of f(x) to OBTAIN g(x).
Can you help me with this?
What I understand is that you want to write/list g(x) in the terms of f(x) and not f(x+1).
Now as we have got f(x+1) = x^3 + 3x^2 + 3x +1, we will rewrite it in terms of f(x).
We know that f(x) = x^3-2x^2+x. Hence we will rearrange f(x+1) to include terms x^3-2x^2+x in its equation as given below:
f(x+1) = x^3 + 3x^2 + 3x +1
= x^3 + 5x^2-2x^2 + x+2x + 1
As you can see here, the equation remains the same but we have broken it down to include terms of f(x). We can rewrite it as,
f(x+1) = x^3-2x^2+x +5x^2+2x+1
= f(x) + 5x^2+2x+1
Thus we get,
f(x+1) = f(x) + 5x^2+2x+1 ...........................(1)
We will write f(x+1) in terms of f(x) as derived in equation (1) above.
g(x) = 3[f(x) + 5x^2+2x+1] - 2
= 3f(x) + 15x^2 + 6x +3 -2
= 3f(x) + 15x^2 + 6x + 1 ...................... (2)
I hope that this is what you are looking for. Please let me know if you need additional information.