The problem asks us to find the equation for the reciprocal function of the parabola graphed in the image. The possible answers are: $f(x) = \frac{1}{x^2+5}$ $f(x) = \frac{1}{x^2-5}$ $f(x) = x^2+5$ $f(x) = x^2-5$

1. Problem Description

The problem asks us to find the equation for the reciprocal function of the parabola graphed in the image. The possible answers are:

$f(x) = \frac{1}{x^2+5}$

$f(x) = \frac{1}{x^2-5}$

$f(x) = x^2+5$

$f(x) = x^2-5$

2. Solution Steps

First, we need to find the equation of the original parabola. The vertex of the parabola is at $(0,-5)$. Therefore, the equation of the parabola is of the form $f(x) = a(x-0)^2 - 5 = ax^2 - 5$. We can see from the graph that when $x=1$, $f(x) = -4$. Plugging this into the equation we have:

$-4 = a(1)^2 - 5$

$-4 = a - 5$

$a = 1$

Therefore, the equation of the original parabola is $f(x) = x^2 - 5$.

The reciprocal of this function is $\frac{1}{f(x)} = \frac{1}{x^2 - 5}$.

3. Final Answer

$f(x) = \frac{1}{x^2-5}$