Question

How can I do this with the function (x+2)/x-3

Answer 1

The given rational function is
$f(x) = \frac{x+2}{x-3}$

Part 1
The horizontal asymptote is obtained by either long division or synthetic division. It may be obtained also as
$f(x)= \frac{x-3+5}{x-3} = \frac{x-3}{x-3} + \frac{5}{x-3} =1+ \frac{5}{x-3}$
Therefore the horizontal asymptote is
y = 1.

The vertical asymptote occurs when the denominator is zero because the function becomes undefined. Set x-3 = 0 to obtain
x = 3.
Therefore a vertical asymptote occurs at x = 3.

The x-intercept occurs when f(x) = y = 0. Set f(x)=0 to obtain
$\frac{x+2}{x-3} =0$
For x≠3, obtain
x+2=0  => x = -2
The x-intercept is x = -2.

The y-intercept occurs when x=0. Set x=0 in f(x) to obtain
$f(0)= \frac{0+2}{0-3} =- \frac{2}{3}$
The y-intercept is
y = -2/3

Part 2
The graph of the function is shown below. It identifies the horizontal and vertical asymptotes, the x-intercept, and the y-intercept.