Question

Consider the following functions.f(x) = x2 + 2x, g(x) = 3x2 - 1Find the domain of (f/g) (x). Enter your answer using interval notation

Answer 1

`\begin{gathered} f(x)=x^2+2x \\ g(x)=3x^2-1 \end{gathered}`

Computing (f/g)(x), we get:

`((f)/(g))(x)=(x^2+2x)/(3x^2-1)`

The denominator can't be zero, that is,

`3x^2-1\\e0`

Solving for x:

`\begin{gathered} 3x^2\\e0+1 \\ x^2\\e(1)/(3) \\ x^{}\\e\sqrt[]{(1)/(3)} \\ \text{This square root has 2 solutions:} \\ x^{}_1\\e\sqrt[]{(1)/(3)} \\ x^{}_2\\e-\sqrt[]{(1)/(3)} \end{gathered}`

Then, the domain of (f/g)(x) is all real values except x1 and x2. In interval notation:

`(-\infty,-\sqrt[]{(1)/(3)})\cup(-\sqrt[]{(1)/(3)},\sqrt[]{(1)/(3)})\cup(\sqrt[]{(1)/(3)},\infty)`