Question

In Problems 13–24, find the domain of each rational function.

$R\left(x\right)=\frac{3(x^2-x-6)}{4(x^2-9)}$

Short answer

The domain of the rational function $R\left(x\right)=\frac{3(x^2-x-6)}{4(x^2-9)}$ is $\{x\overline{)x\ne -3,x\ne 3\}}$

Step-by-step solution

Step 1. Given

The rational function $R\left(x\right)=\frac{3(x^2-x-6)}{4(x^2-9)}$

To find the domain of the given rational function.

Step 2. Factor the denominator.

Factorize the denominator.

$$\begin{gathered} 4(x^2-9)=4(x^2-3^2) \\ =4(x+3)(x-3) \end{gathered}$$

Step 3. Use the zero-Product property

By using Zero-Product property,

$(x+3)(x-3)=0$

Either $x+3=0$or

$x-3=0$

Step 4. Solve the equation

Solve the equations

$$\begin{gathered} x+3=0 \\ x=-3 \end{gathered}$$

And $$\begin{gathered} x-3=0 \\ x=3 \end{gathered}$$

The domain of the function is $\{x\overline{)x\ne -3,x\ne 3\}}$