question 1 of 10, step 2 of 2\nconsider the following functions.\n$f(x) = -2x$ and $g(x) = x - 3$\nstep 2 of…

question 1 of 10, step 2 of 2\nconsider the following functions.\n$f(x) = -2x$ and $g(x) = x - 3$\nstep 2 of 2: find the formula for $left(\\frac{f}{g}\\right)(x)$ and simplify your answer. then find the domain for $left(\\frac{f}{g}\\right)(x)$. round your answer to two decimal places, if necessary.

question 1 of 10, step 2 of 2\nconsider the following functions.\n$f(x) = -2x$ and $g(x) = x - 3$\nstep 2 of 2: find the formula for $left(\\frac{f}{g}\\right)(x)$ and simplify your answer. then find the domain for $left(\\frac{f}{g}\\right)(x)$. round your answer to two decimal places, if necessary.

Answer

Explanation:

Step1: Define the quotient function

$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$ Substitute $f(x)=-2x$ and $g(x)=x-3$: $\left(\frac{f}{g}\right)(x) = \frac{-2x}{x-3}$

Step2: Simplify the expression

The expression $\frac{-2x}{x-3}$ is already in simplest form, or can be rewritten as: $\left(\frac{f}{g}\right)(x) = -\frac{2x}{x-3}$

Step3: Find the domain

The denominator cannot equal 0, so solve $x-3 \neq 0$: $x \neq 3$ In interval notation, this is $(-\infty, 3) \cup (3, \infty)$

Answer:

Formula: $\boldsymbol{\left(\frac{f}{g}\right)(x) = -\frac{2x}{x-3}}$ Domain: All real numbers except $x=3$, or in interval notation $\boldsymbol{(-\infty, 3) \cup (3, \infty)}$