Question

characteristics of rational functions
determine the domain and vertical asymptote for each of the following functions. write the domain as a complete sentence. for example, write the domain as \all real numbers except x = 4\. write the vertical asymptote as an equation.
rational function domain vertical asymptote
$f(x)=\frac{-9}{-12 + 4x}$
$g(x)=\frac{7x + 4}{3x + 6}$
$f(x)=\frac{-7x + 1}{-2x - 8}$
$p(t)=\frac{t}{-6t - 6}$
question help: video message instructor

Explanation:

Step1: Recall domain - denominator non - zero

For a rational function $\frac{a}{b}$, the domain is all real numbers except where $b = 0$. The vertical asymptote is the value of $x$ (or $t$) that makes the denominator zero.

Step2: For $f(x)=\frac{-9}{-12 + 4x}$

Set the denominator $-12 + 4x=0$. Solving for $x$ gives $4x=12$, so $x = 3$.
Domain: All real numbers except $x = 3$.
Vertical Asymptote: $x = 3$.

Step3: For $g(x)=\frac{7x + 4}{3x+6}$

Set the denominator $3x + 6=0$. Solving for $x$ gives $3x=-6$, so $x=-2$.
Domain: All real numbers except $x=-2$.
Vertical Asymptote: $x=-2$.

Step4: For $f(x)=\frac{-7x + 1}{-2x - 8}$

Set the denominator $-2x - 8=0$. Solving for $x$ gives $-2x=8$, so $x=-4$.
Domain: All real numbers except $x=-4$.
Vertical Asymptote: $x=-4$.

Step5: For $p(t)=\frac{t}{-6t - 6}$

Set the denominator $-6t - 6=0$. Solving for $t$ gives $-6t=6$, so $t=-1$.
Domain: All real numbers except $t=-1$.
Vertical Asymptote: $t=-1$.

Answer:

Rational Function	Domain	Vertical Asymptote
$g(x)=\frac{7x + 4}{3x+6}$	All real numbers except $x=-2$	$x=-2$
$f(x)=\frac{-7x + 1}{-2x - 8}$	All real numbers except $x=-4$	$x=-4$
$p(t)=\frac{t}{-6t - 6}$	All real numbers except $t=-1$	$t=-1$