Question

Determine the function which corresponds to the given graph The asymptote is x=3

Answer 1

The function has a horizontal asymptote at y = 3.
The function has a vertical asymptote at x = -3.
The function has a  vertical intercept of 1.
The function has a zero of x = -2.

Because the function has a zero at x = -2, and a vertical asymptote at x = -3, therefore
$y= \frac{k(x+2)}{x+3}$

Write the function in the form
y = $y= \frac{k(1+ \frac{2}{x}) }{1+ \frac{3}{x} }$

As x → ∞, y → k.
Because a horizontal asymptote exists at y = 3, therefore  k = 3.
A vertical intercept exists at y = 1.
The function is
 $y = \frac{3(x+2)(x+1)}{(x+3)(x+1)}$

Answer: $y = \frac{3(x+2)(x+1)}{(x+3)(x+1)}$