Question

find the domain of the following rational function.$f(x)=\frac{3x(x-3)}{3x^2 - 14x - 5}$select the correct choice below and fill in any answer boxes within your choice.○ a. the domain of $f(x)$ is $\\{x \mid \square\\}$.(type an inequality in the form $x \
eq$. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)○ b. the domain of $f(x)$ is the set of all real numbers.

Explanation:

Step1: Set denominator equal to 0

$3x^2 - 14x - 5 = 0$

Step2: Solve quadratic equation

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ for $ax^2+bx+c=0$. Here $a=3$, $b=-14$, $c=-5$:

$$\begin{align*} x&=\frac{14\pm\sqrt{(-14)^2-4(3)(-5)}}{2(3)}\\ &=\frac{14\pm\sqrt{196+60}}{6}\\ &=\frac{14\pm\sqrt{256}}{6}\\ &=\frac{14\pm16}{6} \end{align*}$$

Calculate roots:
$x=\frac{14+16}{6}=\frac{30}{6}=5$, $x=\frac{14-16}{6}=\frac{-2}{6}=-\frac{1}{3}$

Step3: Exclude roots from domain

Domain excludes $x=5$ and $x=-\frac{1}{3}$

Answer:

A. The domain of $F(x)$ is $\{x \mid x
eq -\frac{1}{3}, 5\}$