If f(x)= 3x+5 and h(x)= 3x^2 +3x+2, find a function g such that f o g=h .

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`f(x) = 3x+5`

`h(x) = 3x^2 +3x+2`

In `f_og` the term x in function f(x) is replace by g(x).

`f_0g = 3g(X)+5`

`f_0g = h(x)`

`3g(x)+5 = 3x^2 +3x+2`

`3g(x) =3x^2 +3x-3 `

`g(x) = x^2+x-1`

**So the function `g(x) = x^2+x-1` **

Just by trial and error, it can be seen that if `g(x)=x^2+x-1`, then

`(f@g)(x)=f(g(x))=f(x^2+x-1)=3(x^2+x-1)+5`

`=3x^2+3x-3+5``=3x^2+3x+2=h(x),`

` ` so that's the answer. There is a way to do it without having to guess, though. Note that if we have

`f@g=h,` then `g=(f^(-1)@f)@g=f^(-1)@(f@g)=f^(-1)@h.`

Since `f^(-1)(x)=(x-5)/3,`

`(f^(-1)@h)(x)=f^(-1)(3x^2+3x+2)=((3x^2+3x-2)+5)/3`

`=x^2+x-1.`

**So either way we do it, we get `g(x)=x^2+x-1.` **

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