Question

Uf(x)=2x2−5x+3

g(x)=4x2−12x+9

Find: (fg)(x)

Answer 1

Your final answer will be: 16x^4 - 80x^3 + 138x^2 - 95x + 24

I will assume that the first equation is:

f(x) = 2x^2 - 5x + 3

and the second equation is:

4x^2 - 12x + 09.

To find f(g(x)) you need to plug in the equation of g(x) as x into the equation of f(x). So you get:

4(2x^2 - 5x + 3)^2 - 5(2x^2 - 5x + 3) + 3.

That then equals to:

4(4x^4 - 20x^3 + 37x^2 - 30x+9) - 5(2x^2 - 5x + 3) + 3.

Further simplified you get:

16x^4 - 80x^3 + 148x^2 - 120x + 36 - 10x^2 + 25x - 15 + 3.

That simplifies to:

16x^4 - 80x^3 + 138x^2 - 95x + 24.

Answer 2

`(fg)(x)= 16x^4 - 80x^3 + 124x^2 -60x +9`

Explanation:

In general, the composition of fuction is :

(fg)(x) = f(g(x))

In our case, we do the next:

`f(x) = 2x^2-5x+3`

`g(x) = 4x^2-12x+9`

and the composition is

`4(2x^2-5x+3)^2-12(2x^2-5x+3)+9`

The first term is:

`2x^2-5x+3 = (2x^2-5x+3)(2x^2-5x+3) = 4x^4-20x^3+37x^2-30x+9`

⇒
`4(4x^4-20x^3+37x^2-30x+9)-12(2x^2-5x+3)+9`

⇒
`16x^4-80x^3+148x^2-120x+36-12\left(2x^2-5x+3\right)+9`

⇒
`16x^4-80x^3+148x^2-120x+36-24x^2+60x-36+9`

Simplifying, we have

⇒
`16x^4-80x^3+124x^2-60x+9`