Question

a. $f(x) = \frac{(x + 3)(x - 4)}{-4(x - 3)}$
b. $f(x) = \frac{4(x - 3)}{(x + 3)(x - 4)}$
c. $f(x) = \frac{(x + 3)(x - 4)}{4(x - 3)}$
d. $f(x) = \frac{(x - 3)}{4(x + 3)(x - 4)}$

Explanation:

Step1: Identify vertical asymptotes

Vertical asymptotes occur where the denominator is 0 (and numerator is not 0 at those points). From the graph, vertical asymptotes are at $x=-3$ and $x=4$. So the denominator must have factors $(x+3)$ and $(x-4)$. This eliminates options a and c.

Step2: Identify x-intercept

The x-intercept is where $f(x)=0$, so numerator is 0. The graph crosses the x-axis at $x=3$, so the numerator must have a factor $(x-3)$. This eliminates option d.

Step3: Verify sign behavior

For $x>4$, the graph is positive. For option b, substitute $x=5$: $f(5)=\frac{4(5-3)}{(5+3)(5-4)}=\frac{8}{8}=1>0$, which matches the graph. For $00$, which matches.

Answer:

b. $f(x) = \frac{4(x - 3)}{(x + 3)(x - 4)}$