Question

If f(x)=2x+1 and g(f(x))=4x^2+4x+3 find g(x) given that g(x)=ax^2+bx+c how do I do that?

I know that somebody has solved this before but I would like a better version of that one since I did not understand what was going on in the other one. Many thanks!

Answer 1

`g(x) = x^(2) + 2`

Explanation:

We know that `f(x) = 2 x + 1` and `g(x) = a x^(2) + b x + c`.

`Rightarrow g(f(x)) = a (2 x + 1)^(2) + b (2 x + 1) + c`

Expanding the parentheses:

`Rightarrow g(f(x)) = a (4 x^(2) + 4 x + 1) + 2 b x + b + c`

`Rightarrow g(f(x)) = 4 a x^(2) + 4 a x + a + 2 b x + b + c`

Rearranging the variables:

`Rightarrow g(f(x)) = 4 a x^(2) + (4 a + 2 b) x + a + b + c`

Now, we are also given the fact that `g(f(x)) = 4 x^(2) + 4 x + 3`.

Comparing coefficients:

`Rightarrow 4 a = 4 therefore a = 1`

`and`

`Rightarrow 4 a + 2 b = 4 Rightarrow 4 times (1) + 2 b = 4 Rightarrow4 + 2 b = 4 Rightarrow 2 b = 0 therefore b = 0`

`and`

`Rightarrow a + b + c = 3 Rightarrow (1) + (0) + c = 3 Rightarrow 1 + c = 3 therefore c = 2`

Therefore, `g(x) = (1) x^(2) + (0) x + (2) = x^(2) + 2`.