Question

How do you find the compositions given `f(x)=3x^2-6x+5` and `g(x)=x^2+5x-2`?

Answer 1

I help you find one and leave the rest up to you.

How do you find the composition `(f @g)(x)`? After obtaining that answer, determine the numerical value of `f(g(-2))`.

First note that `(f@g)(x) = f(g(x))`. This means to insert `g(x)` into `f(x)`.

`f(g(x)) = 3(x^2 + 5x - 2)^2 - 6(x^2 + 5x - 2) + 5`

Doing the algebra, we have:

`f(g(x)) = 3x^4 + 30x^3 + 57x^2 - 90x + 29`

We obtain `f(g(-2))` by inserting `x = -2` within `f(g(x))`.

`f(g(-2)) = 3(-2)^4 + 30(-2)^3 + 57(-2)^2 - 90(-2) + 29`

`f(g(-2)) = 3(16) + 30(-8) + 57(4) + 90(2) + 29`

`f(g(-2)) = 245`

Hopefully this helps!