Question

Write the function , √(x^3+6)/√(x^3-9) as a composition of three or more non-identity functions.

Answer 1

`h \circ m \circ n \text{ where } h(x)=√(x) \text{ and } m(x)=1+(15)/(n) \text{ and } n(x)=x^3-9`

Explanation:

Ok so I see a square root is on the whole thing.

I'm going to let the very outside function by
`h(x)=sqrt(x)`.

Now I'm can't just let the inside function by one function
`g(x)=(x^3+6)/(x^3-9)` because we need three functions.

So I'm going to play with
`g(x)=(x^3+6)/(x^3-9)` a little to simplify it.

You could do long division. I'm just going to rewrite the top as

`x^3+6=x^3-9+15`.

`g(x)=(x^3-9+15)/(x^3-9)=1+(15)/(x^3-9)`.

So I'm going to let the next inside function after h be
`m(x)=1 + (15)/(x)`.

Now my last function will be
`n(x)=x^3-9`.

So my order is h(m(n(x))).

Let's check it:

`h(m(x^3-9))`

`h(1+(15)/(x^3-9))`

`h((x^3-9+15)/(x^3-9))`

`h((x^3+6)/(x^3-9))`

`\sqrt{ (x^3+6)/(x^3-9)}`