Question

How do you find (f of g of h) if `f(x)=x^2+1` `g(x)=2x` and `h(x)=x-1`?

Answer 1

`(f@g@h)(x) = 4x^2-8x+5`

Explanation:

Given:

`{ (f(x) = x^2+1), (g(x) = 2x), (h(x) = x-1) :}`

One way of thinking about these function compositions is to go back and forth between the symbols and verbal descriptions of what the functions do.

In our example:

• `f` takes the square of a number and adds `1`

• `g` doubles a number

• `h` subtracts `1` from a number

So a verbal description of the composed `f@g@h` as a sequence of steps might be:

• Subtract `1`

• Double

• Square

• Add `1`

So in symbols we might describe this process thus:

`x -> x-1 -> 2(x-1) -> (2(x-1))^2 -> (2(x-1))^2+1`

So:

`(f@g@h)(x) = f(g(h(x)))`

`color(white)((f@g@h)(x)) = (2(x-1))^2+1`

`color(white)((f@g@h)(x)) = 4(x^2-2x+1)+1`

`color(white)((f@g@h)(x)) = 4x^2-8x+4+1`

`color(white)((f@g@h)(x)) = 4x^2-8x+5`