Rational Function: Find Intervals Where R(x) ≥ 0

Hey Plastik Magazine readers! Today, we're diving into the fascinating world of rational functions. Specifically, we're tackling a problem where we need to figure out the intervals of x for which a given rational function, r(x), is greater than or equal to zero. Sounds like fun, right? Let's break it down step by step.

Understanding Rational Functions

Before we jump into the problem, let's quickly recap what rational functions are all about. A rational function is essentially a fraction where both the numerator and the denominator are polynomials. Our function today is r(x) = (x^3 + 4x^2 + 4x) / (x^2 - 9). The key to solving inequalities involving rational functions lies in understanding their behavior, especially around points where the function might change its sign – these are the zeros (where the numerator is zero) and the points of discontinuity (where the denominator is zero).

Step 1: Factor Everything!

The first crucial step in solving this problem is to factor both the numerator and the denominator as much as possible. Factoring helps us identify the critical points where the function might change its sign. For the numerator, x^3 + 4x^2 + 4x, we can factor out an x first, giving us x(x^2 + 4x + 4). The quadratic part, x^2 + 4x + 4, is a perfect square trinomial, which factors to (x + 2)^2. So, the numerator becomes x(x + 2)^2.

Now, let's tackle the denominator, x^2 - 9. This is a classic difference of squares, which factors to (x - 3)(x + 3). Putting it all together, our rational function looks like this: r(x) = x(x + 2)^2 / ((x - 3)(x + 3)). See how much clearer things become when we factor? We can now easily spot the values of x that make the numerator or the denominator zero.

Step 2: Identify Critical Points

The critical points are the values of x where the function r(x) could potentially change its sign. These occur when the numerator is zero (giving us the zeros of the function) or when the denominator is zero (giving us vertical asymptotes, where the function is undefined). From the factored form, r(x) = x(x + 2)^2 / ((x - 3)(x + 3)), we can identify the following critical points:

• x = 0: This comes from the x term in the numerator.
• x = -2: This comes from the (x + 2)^2 term in the numerator. Notice that this term is squared, which will be important later when we consider the sign changes.
• x = 3: This comes from the (x - 3) term in the denominator.
• x = -3: This comes from the (x + 3) term in the denominator.

These four critical points, -3, -2, 0, and 3, divide the number line into five intervals. Our next task is to determine the sign of r(x) in each of these intervals.

Step 3: Create a Sign Chart

A sign chart is a fantastic tool for visualizing the sign of a function over different intervals. We'll create a table with the critical points as dividers and analyze the sign of each factor in each interval.

First, draw a number line and mark the critical points: -3, -2, 0, and 3. These points divide the number line into the following intervals:

1. (-∞, -3)
2. (-3, -2)
3. (-2, 0)
4. (0, 3)
5. (3, ∞)

Now, let's create a table and analyze the sign of each factor in each interval:

Let's break down how we filled in this table. For each interval, we pick a test value within that interval and determine the sign of each factor. For example, in the interval (-∞, -3), we could pick x = -4. Plugging this into each factor:

• x = -4 is negative.
• (x + 2)^2 = (-4 + 2)^2 = 4 is positive (since it's squared).
• x - 3 = -4 - 3 = -7 is negative.
• x + 3 = -4 + 3 = -1 is negative.

The sign of r(x) is the product of the signs of the factors. In this case, (-)(+)(-) / (-) which simplifies to negative. We repeat this process for each interval.

Step 4: Determine the Intervals Where r(x) ≥ 0

We're looking for intervals where r(x) is greater than or equal to 0. From our sign chart, we can see that r(x) is positive in the intervals (-3, -2) and (-2, 0) and (3, ∞). It's also equal to zero at x = 0 and x = -2. We need to include these points because the question asks for r(x) ≥ 0. However, we must exclude x = -3 and x = 3 because the function is undefined at these points (the denominator is zero).

Therefore, the intervals where r(x) ≥ 0 are: (-3, -2] ∪ [-2, 0] ∪ (3, ∞).

Notice how we use parentheses for -3 and 3 (because they are not included) and square brackets for -2 and 0 (because they are included). The symbol ∪ represents the union of these intervals.

Final Answer

The intervals of x where the rational function r(x) = (x^3 + 4x^2 + 4x) / (x^2 - 9) is greater than or equal to 0 are: (-3, -2] ∪ [-2, 0] ∪ (3, ∞).

Key Takeaways

Let's recap the key steps to solve inequalities involving rational functions:

1. Factor: Factor the numerator and denominator completely.
2. Identify Critical Points: Find the zeros of the numerator and the denominator.
3. Create a Sign Chart: Analyze the sign of each factor in each interval created by the critical points.
4. Determine Intervals: Identify the intervals where the function satisfies the inequality, remembering to include or exclude endpoints based on the inequality symbol and the function's domain.

Understanding rational functions and mastering these steps can help you tackle a wide range of mathematical problems. Keep practicing, and you'll become a pro in no time! Solving rational inequalities involves a systematic approach, and hopefully, this breakdown helps you feel more confident in tackling similar problems. Remember, the key is to break down the problem into smaller, manageable steps. Factoring, identifying critical points, and using a sign chart are your best friends in this process.

So, there you have it, guys! A comprehensive guide to finding intervals where a rational function is greater than or equal to zero. Until next time, keep exploring the fascinating world of mathematics!