Question

How do you find the compositions given `g(x) = 3x + 2` and `h(x) = 9x^2 + 12x + 9`?

Answer 1

`g(h(x))=27x^2+36x+29`
`h(g(x))=81x^2+144x+69`

Explanation:

Function compositions are basically plugging one function into another function.

To find `g(h(x))`, take `h(x)`, which is `9x^2+12+9`, and plug it into the `x` in `g(x)`.

`g(color(blue)(x))=3color(blue)x+2`
`h(x)=9x^2+12x+9`

`g(color(blue)(h(x)))=3color(blue)((9x^2+12x+9))+2`
`g(h(x))=27x^2+36x+27+2`
`g(h(x))=27x^2+36x+29`

To find `h(g(x))`, do the process with the roles switched: `g(x)` is plugged into `h(x)`.

`h(g(x))=9(3x+2)^2+12(3x+2)+9`
`h(g(x))=9(9x^2+12x+4)+36x+24+9`
`h(g(x))=81x^2+108x+36+36x+33`
`h(g(x))=81x^2+144x+69`