Question

use the graphing tool to graph function f.
$f(x) = \frac{x^2 - 5x - 24}{x - 8}$
the graph of function f has asymptote, and the range is .

Explanation:

Step1: Simplify the rational function

First, factor the numerator:
$x^2 - 5x - 24 = (x - 8)(x + 3)$
So $f(x) = \frac{(x - 8)(x + 3)}{x - 8} = x + 3$, for $x
eq 8$

Step2: Identify asymptotes

Since the factor $(x-8)$ cancels, there is no vertical asymptote (only a hole at $x=8$). The simplified function is linear, so no horizontal/oblique asymptote other than the line itself, but the hole means the graph has no asymptote.

Step3: Determine the range

The simplified function $y = x + 3$ has a range of all real numbers, but since $x
eq 8$, $y
eq 8 + 3 = 11$. So the range is all real numbers except 11.

Answer:

The graph of function $f$ has no asymptote, and the range is all real numbers except 11 (or in set notation: $(-\infty, 11) \cup (11, \infty)$)