Question

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

Answer 1

Answer:

$h \circ m \circ n \text{ where } h(x)=\sqrt{x} \text{ and } m(x)=1+\frac{15}{n} \text{ and } n(x)=x^3-9$

Step-by-step 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 $g(x)=\frac{x^3+6}{x^3-9}$ because we need three functions.

So I'm going to play with $g(x)=\frac{x^3+6}{x^3-9}$ a little to simplify it.

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

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

$g(x)=\frac{x^3-9+15}{x^3-9}=1+\frac{15}{x^3-9}$.

So I'm going to let the next inside function after h be $m(x)=1 + \frac{15}{x}$.

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

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

Let's check it:

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

$h(1+\frac{15}{x^3-9})$

$h(\frac{x^3-9+15}{x^3-9})$

$h(\frac{x^3+6}{x^3-9})$

$\sqrt{ \frac{x^3+6}{x^3-9}}$