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:

$(\frac{f}{g})(x)=\frac{x^2+2x}{3x^2-1}$

The denominator can't be zero, that is,

$3x^2-1\ne0$

Solving for x:

$\begin{gathered} 3x^2\ne0+1 \\ x^2\ne\frac{1}{3} \\ x^{}\ne\sqrt[]{\frac{1}{3}} \\ \text{This square root has 2 solutions:} \\ x^{}_1\ne\sqrt[]{\frac{1}{3}} \\ x^{}_2\ne-\sqrt[]{\frac{1}{3}} \end{gathered}$

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

$(-\infty,-\sqrt[]{\frac{1}{3}})\cup(-\sqrt[]{\frac{1}{3}},\sqrt[]{\frac{1}{3}})\cup(\sqrt[]{\frac{1}{3}},\infty)$