Analyzing Function Behavior: Increasing, Decreasing, And Constant Intervals

Hey math enthusiasts! Today, we're diving into the fascinating world of function behavior. Specifically, we'll learn how to identify the open interval(s) where a function is either increasing, decreasing, or constant. This is super important because it helps us understand how a function changes its values as its input changes. We'll be working with the function $p(x) = \frac{2}{x+3}$, and our goal is to express our findings in interval notation. So, grab your pencils, and let's get started!

Understanding Increasing, Decreasing, and Constant Functions

Before we jump into the problem, let's quickly recap what it means for a function to be increasing, decreasing, or constant. These concepts are the bedrock of our analysis, so it's vital to have a solid grasp of them.

• Increasing Function: A function is increasing on an interval if, as the x-values (inputs) increase, the y-values (outputs) also increase. Graphically, this means the function's curve is going uphill as you move from left to right. Think of it like climbing a mountain – your height (y-value) increases as you move along the path (x-value).
• Decreasing Function: Conversely, a function is decreasing on an interval if, as the x-values increase, the y-values decrease. On a graph, this looks like the function's curve going downhill as you move from left to right. Imagine descending a mountain – your height (y-value) decreases as you move forward (x-value).
• Constant Function: Finally, a function is constant on an interval if its y-values remain unchanged, regardless of the x-value. The graph of a constant function is a horizontal line. It's like walking on a perfectly flat surface; your height (y-value) stays the same no matter how far you walk (x-value).

Now, let's apply these definitions to our specific function, $p(x) = \frac{2}{x+3}$. Our primary task is to find the intervals where $p(x)$ behaves in any of these three ways. We will carefully examine the function's behavior to determine where it increases, decreases, or remains constant. This analysis will involve finding the critical points, and the intervals between these points. Understanding these behaviors is fundamental to calculus and helps us understand complex functions, helping you become a math whiz. The goal is to provide a complete understanding of the function's behavior, which is a critical skill in calculus and other areas of mathematics. So, let's gear up and find the solution.

Analyzing the Function $p(x) = \frac{2}{x+3}$

Alright, let's get our hands dirty and analyze the function $p(x) = \frac{2}{x+3}$. The first thing we need to do is to determine the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, we have a rational function (a fraction), so we must consider any values of x that would make the denominator equal to zero, because division by zero is undefined.

The denominator is $x + 3$. To find any points where the function is undefined, we set the denominator equal to zero and solve for x:

$x + 3 = 0$ $x = -3$

So, the function is undefined at $x = -3$. This means our domain will be all real numbers except -3. We can represent this in interval notation as $(-\infty, -3) \cup (-3, \infty)$. This is a crucial point because it separates the x-axis into two distinct intervals, and the behavior of the function may be different on each side of $x = -3$.

Now, let's try to determine where the function is increasing, decreasing, or constant. We can do this by using the first derivative test. To use the first derivative test, we'll first need to find the derivative of $p(x)$.

$p(x) = \frac{2}{x+3} = 2(x+3)^{-1}$

Using the chain rule, we can find the derivative $p'(x)$:

$p'(x) = -2(x+3)^{-2} = \frac{-2}{(x+3)^2}$

The first derivative $p'(x)$ tells us about the slope of the original function $p(x)$. Specifically:

• If $p'(x) > 0$, then $p(x)$ is increasing.
• If $p'(x) < 0$, then $p(x)$ is decreasing.
• If $p'(x) = 0$, then $p(x)$ is constant or has a critical point.

Looking at our derivative $p'(x) = \frac{-2}{(x+3)^2}$, we can see a couple of key things. The numerator is -2, which is always negative. The denominator is $(x+3)^2$, which is always positive or zero (since any real number squared is non-negative). However, remember that $x \neq -3$ due to the domain restriction.

Therefore, the denominator will never be zero, and since the numerator is negative and the denominator is always positive, the fraction $\frac{-2}{(x+3)^2}$ is always negative for all $x$ in the domain of the function. This means that $p'(x) < 0$ for all $x$ except $x = -3$. This tells us that the function is decreasing everywhere in its domain. The function is always decreasing in the interval. The function is not increasing on any interval. Because the derivative is never zero, there are no intervals where the function is constant.

Determining the Intervals

Let's summarize our findings to determine the intervals. We've established that the function is decreasing everywhere in its domain.

• Increasing: None
• Decreasing: $(-\infty, -3) \cup (-3, \infty)$
• Constant: None

Therefore, based on our analysis, the correct answer is B. Decreasing on one interval. This detailed analysis should provide you with a clearer understanding of how to determine the increasing, decreasing, and constant intervals of a given function using calculus principles. The use of derivatives allows us to identify the function's rate of change, thus providing us with insights on its behavior.

Conclusion

Alright, guys, we've successfully navigated the analysis of the function $p(x) = \frac{2}{x+3}$ and determined its increasing, decreasing, and constant intervals. Remember, the key is to understand the relationship between the first derivative and the function's behavior. We first found the derivative, looked at the sign of the derivative, and used that information to figure out whether the function was increasing, decreasing, or constant. This is a fundamental concept in calculus and is super useful for understanding and graphing functions.

So, keep practicing, keep exploring, and keep the math vibes flowing! Understanding these concepts will give you a solid foundation for tackling more complex mathematical problems. Now you are well-equipped to analyze other functions. Keep practicing to become a math whiz. The more you practice, the easier it becomes. Happy learning!