Question

How do you find the compositions given `f(x)=-3x+5`, `g(x)=1+2x-x^2`?

Answer 1

There are many compositions possible. I'll give you one: `ƒ(g(x))`

Explanation:

`ƒ(g(x))` means to plug function g into function ƒ. The function on the inside is always to be plugged into the function on the outside. This can also be notated `(ƒ @ g)(x)`, which is read inside to outside as well. When a number is written in place of x, it means to use the number as the value of x in the inner function, which will give you a numerical result. Afterwards, you plug that number into x in the outer function.

`ƒ(g(x))` = `ƒ(-x^2 + 2x + 1)`

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

=`3x^2 - 6x - 3 + 5`

=`3x^2 - 6x + 2`

So, `ƒ(g(x)) = 3x^2 - 6x + 2`

Practice exercises:

1. Assuming `ƒ(x) = 1/(3x - 4)` and `g(x) = 3x^2 - 6`.

Find the following compositions:

a) `g(ƒ(x))`

b) `ƒ(g(-4))`

Good luck!