Question

Let g(x)=x^2-64 and h(x)=root(8x+512). Find each combination and state its domain. 1. (h/g)(x)(Number one has a fraction) 2. (h・g)(x)3. (g・h)(x)Show work please!

Answer 1

Given the functions:

`\begin{gathered} g(x)=x^2-64 \\ h(x)=\sqrt[]{8x+512} \end{gathered}`

1. (h/g)(x)

`((h)/(g))(x)=(h(x))/(g(x)),\text{ g(x)}\\e0`

We substitute the functions and solve:

`((h)/(g))(x)=\frac{\sqrt[]{8x+512}}{x^2-64}`

Factor the common term 8:

`=\frac{\sqrt[]{8(x+64)}}{x^2-64}`

And we have:

`\begin{gathered} x^2-64\\e0 \\ x^2-64+64\\e0+64 \\ x^2\\e64 \\ x\\e8\text{ and x}\\e-8 \end{gathered}`