a rational function f(x) is shown plotted on the graph below. there are no x - intercepts outside the…

Explanation:

Step1: Identify vertical asymptotes

The vertical asymptotes are at $x=-4$ and $x = 2$. So the denominator has factors of $(x + 4)$ and $(x - 2)$.

Step2: Identify x - intercept

The x - intercept is at $x=-2$, so the numerator has a factor of $(x + 2)$.

Step3: Identify y - intercept

The y - intercept is at $(0,2)$. Let the rational function be $f(x)=\frac{a(x + 2)}{(x + 4)(x - 2)}$. Substitute $x = 0$ and $y=2$ into the function: $2=\frac{a(0 + 2)}{(0 + 4)(0 - 2)}$, $2=\frac{2a}{-8}$, $-16 = 2a$, $a=-8$.

Answer:

$f(x)=\frac{-8(x + 2)}{(x + 4)(x - 2)}$