Question3. The graph of which of the following functions in the $x y$-plane has at least one $x$-intercept, at least one hole, at least one vertical isf motote, and a horizontal asymptote. $$f(x)=\frac{x^{2}-1}{x^{2}-x-30}$$ Q. $$f(x)=\frac{x^{2}-1}{x^{3}-x-6}$$ 0. $f(x)=\frac{x^{2}-16}{x^{2}-x-6}$ Previous 1 () () 3 ® 4 0 5 (3) 6 () 7 0 8 . 10

Studdy Solution
- $f(x) = \frac{x^2 - 1}{x^2 - x - 30}$ has $x$-intercepts at $x = \pm 1$, no holes, vertical asymptotes at $x = 6$ and $x = -5$, and a horizontal asymptote at $y = 1$. - $f(x) = \frac{x^2 - 1}{x^3 - x - 6}$ has $x$-intercepts at $x = \pm 1$, no holes, vertical asymptotes determined by the roots of $x^3 - x - 6 = 0$, and a horizontal asymptote at $y = 0$. - $f(x) = \frac{x^2 - 16}{x^2 - x - 6}$ has $x$-intercepts at $x = \pm 4$, a hole at $x = 3$, vertical asymptotes at $x = 3$ and $x = -2$, and a horizontal asymptote at $y = 1$.
The function $f(x) = \frac{x^2 - 16}{x^2 - x - 6}$ satisfies all conditions: it has $x$-intercepts, a hole, vertical asymptotes, and a horizontal asymptote.
The function is:
$$\boxed{f(x) = \frac{x^2 - 16}{x^2 - x - 6}}$$