Question

the graph of ( f ) is shown below.
write an equation of the rational function that represents the graph.
( r(x) = \boxed{3 - \frac{2}{x}} \times )

Explanation:

Step1: Identify horizontal asymptote

The horizontal dashed line is $y=3$, so the rational function has a horizontal asymptote at $y=3$. This means the leading terms of numerator and denominator have a ratio of 3.

Step2: Identify vertical asymptote

The vertical asymptote is $x=2$, so the denominator has a factor of $(x-2)$.

Step3: Form basic function structure

Start with $R(x) = 3 + \frac{k}{x-2}$, where $k$ is a constant to find.

Step4: Use a point to solve for $k$

Pick the point $(0,1)$ from the graph:
$1 = 3 + \frac{k}{0-2}$
$1 - 3 = \frac{k}{-2}$
$-2 = \frac{k}{-2}$
$k = 4$

Step5: Simplify the function

Substitute $k=4$ and rewrite:
$R(x) = 3 + \frac{4}{x-2} = \frac{3(x-2) + 4}{x-2} = \frac{3x - 6 + 4}{x-2} = \frac{3x - 2}{x-2}$

Answer:

$R(x)=\frac{3x-2}{x-2}$