Where Is F(x) = (9-x^2)/(x^2-4) Positive?

Hey there, math enthusiasts! Today, we're diving into a fun problem involving rational functions and their signs. We're going to analyze the function $f(x)=\frac{9-x^2}{x^2-4}$ and pinpoint the intervals where it happily lives above the x-axis, meaning $f(x) > 0$. This kind of analysis is super important in calculus and other areas of math, helping us understand the behavior of functions. So, grab your pencils, and let's break this down step-by-step.

Understanding Rational Functions and Their Signs

When we talk about a rational function, we're essentially dealing with a fraction where both the numerator and the denominator are polynomials. The sign of a rational function, whether it's positive or negative, depends on the signs of its numerator and denominator. For $f(x) = \frac{9-x^2}{x^2-4}$ to be positive, two main scenarios can occur:

1. Both the numerator and the denominator are positive. This means $9-x^2 > 0$ AND $x^2-4 > 0$.
2. Both the numerator and the denominator are negative. This means $9-x^2 < 0$ AND $x^2-4 < 0$.

To figure out where these conditions hold true, we first need to find the roots of both the numerator and the denominator. These roots are critical points where the function's sign might change. They divide the number line into different intervals, and within each interval, the sign of $f(x)$ remains constant.

Finding the Critical Points

Let's start with the numerator: $9-x^2$. To find its roots, we set it equal to zero:

$9 - x^2 = 0$ $x^2 = 9$ $x = \pm 3$

So, our critical points from the numerator are $x = -3$ and $x = 3$. These are the points where $f(x)$ could potentially be zero.

Now, let's look at the denominator: $x^2-4$. We set it equal to zero to find its roots:

$x^2 - 4 = 0$ $x^2 = 4$ $x = \pm 2$

Our critical points from the denominator are $x = -2$ and $x = 2$. These are important because they are the values of $x$ where the function is undefined (division by zero!).

Analyzing the Intervals

Our critical points, in increasing order, are -3, -2, 2, and 3. These points divide the number line into five distinct intervals:

• $(-\infty, -3)$
• $(-3, -2)$
• $(-2, 2)$
• $(2, 3)$
• $(3, \infty)$

Now, we need to pick a test value within each interval and plug it into $f(x)$ to determine the sign of the function in that interval. Let's make a table to keep track of our findings. For each interval, we'll evaluate the sign of the numerator ($9-x^2$) and the denominator ($x^2-4$), and then determine the sign of $f(x)$.

Identifying the Intervals Where $f(x)$ is Positive

From our table, we can clearly see that $f(x)$ is positive in the following intervals:

• $(-3, -2)$
• $(2, 3)$

These are the intervals where both the numerator and the denominator are positive. Let's double-check our logic:

• For the numerator $9-x^2$ to be positive, we need $x^2 < 9$, which means $-3 < x < 3$. This covers intervals $(-3, -2)$, $(-2, 2)$, and $(2, 3)$.
• For the denominator $x^2-4$ to be positive, we need $x^2 > 4$, which means $x < -2$ or $x > 2$. This covers intervals $(-\infty, -3)$, $(-3, -2)$, $(2, 3)$, and $(3, \infty)$.

Now, let's find the intersection of these conditions. We need BOTH the numerator AND the denominator to be positive. So, we're looking for the x-values that satisfy both $-3 < x < 3$ AND ($x < -2$ or $x > 2$).

• If $-3 < x < 3$ AND $x < -2$, the overlap is $-3 < x < -2$. This gives us the interval $(-3, -2)$.
• If $-3 < x < 3$ AND $x > 2$, the overlap is $2 < x < 3$. This gives us the interval $(2, 3)$.

Therefore, the intervals where $f(x)$ is positive are indeed $(-3, -2)$ and $(2, 3)$.

Conclusion: Which Intervals are Positive?

Based on our thorough analysis, the function $f(x)=\frac{9-x^2}{x^2-4}$ is positive in the intervals $(-3, -2)$ and $(2, 3)$. These correspond to options B and D in the given choices.

This type of sign analysis is a fundamental skill in understanding function behavior, especially when preparing for calculus concepts like curve sketching and optimization. Remember, the key is to find the roots, define the intervals, and test a value in each interval. It's like being a detective for functions!

For further exploration into the graphical behavior of rational functions and how to determine their signs, you can check out resources like Khan Academy's comprehensive guides on rational functions. They offer excellent explanations and practice problems that can solidify your understanding.