Identifying Decreasing Intervals Of Functions And Representing Them On A Number Line

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Introduction

In the realm of mathematics, understanding the behavior of functions is crucial. One important aspect of function analysis is identifying intervals where a function is increasing or decreasing. This article will delve into the concept of decreasing intervals, particularly focusing on two functions: Function 1: $f(x)=-5|x+1|+10$ and Function 2 (represented graphically). We will explore how to determine where these functions are decreasing and, most importantly, how to represent the interval where both functions are decreasing on a number line. This involves a combination of algebraic analysis, graphical interpretation, and a clear understanding of interval notation. By the end of this article, you'll have a solid grasp of how to identify and represent decreasing intervals, a fundamental skill in calculus and beyond.

Analyzing Function 1: $f(x)=-5|x+1|+10$

To understand where the function $f(x)=-5|x+1|+10$ is decreasing, we first need to analyze its structure. This function is an absolute value function, which means it has a V-shape. The general form of an absolute value function is $f(x) = a|x-h|+k$, where:

• a determines the direction and stretch/compression of the graph.
• (h, k) is the vertex of the V-shape.

In our case, $f(x)=-5|x+1|+10$, we have $a = -5$, $h = -1$, and $k = 10$. The negative value of a tells us that the V-shape is inverted, opening downwards. The vertex of the graph is at the point (-1, 10). This is a crucial piece of information because the vertex is the turning point of the function. To the left of the vertex, the function will be increasing (going upwards), and to the right of the vertex, the function will be decreasing (going downwards).

The absolute value function $|x+1|$ is defined piecewise as follows:

• $|x+1| = x+1$ when $x+1 \geq 0$, which means $x \geq -1$
• $|x+1| = -(x+1)$ when $x+1 < 0$, which means $x < -1$

Therefore, we can rewrite the original function piecewise:

• $f(x) = -5(x+1) + 10 = -5x + 5$ when $x \geq -1$
• $f(x) = -5(-(x+1)) + 10 = 5x + 15$ when $x < -1$

Now we can clearly see the two linear pieces of the function. For $x < -1$, the slope is 5, indicating an increasing function. For $x \geq -1$, the slope is -5, indicating a decreasing function. Consequently, Function 1, $f(x) = -5|x+1| + 10$, is decreasing on the interval $[-1, \infty)$. This is because the function goes downwards as x increases from -1.

Interpreting Function 2 Graphically and Identifying Decreasing Intervals

Function 2 is presented graphically, meaning we don't have an algebraic equation to analyze. Instead, we rely on visual cues to determine where the function is decreasing. A function is decreasing where its graph is going downwards as you move from left to right along the x-axis. This is equivalent to saying that the slope of the function is negative in the decreasing interval.

To identify the decreasing interval(s) of Function 2, we need to carefully examine the graph. Look for sections where the line or curve slopes downwards. Note the x-values at the beginning and end of these downward-sloping sections. These x-values will define the interval(s) where the function is decreasing.

For example, if the graph slopes downwards from x = 2 to x = 5, then the function is decreasing on the interval [2, 5]. The interval notation includes square brackets [ ] if the endpoints are included in the interval and parentheses ( ) if the endpoints are excluded. Endpoints are typically excluded if the graph has a vertical asymptote or an open circle at that point.

Without the specific graph of Function 2, we can only provide a general methodology. However, let's assume, for the sake of illustration, that after examining the graph, we determine that Function 2 is decreasing on the interval (3, 7]. This means the function's values decrease as x moves from slightly greater than 3 (not including 3) up to and including 7.

Determining the Intersection of Decreasing Intervals

The core of this problem lies in finding the interval where both functions are decreasing. This means we need to find the intersection of the decreasing intervals of Function 1 and Function 2. In other words, we need to identify the values of x for which both functions are simultaneously decreasing.

We previously determined that Function 1 is decreasing on the interval $[-1, \infty)$. We also assumed (for illustrative purposes) that Function 2 is decreasing on the interval (3, 7]. To find the intersection of these intervals, we can visualize them on a number line or use algebraic methods.

• Interval 1 (Function 1): $[-1, \infty)$ represents all numbers greater than or equal to -1.
• Interval 2 (Function 2): (3, 7] represents all numbers strictly greater than 3 and less than or equal to 7.

The intersection is the set of numbers that are in both intervals. We can see that the overlapping region starts at 3 (since Function 2's interval excludes values less than or equal to 3) and ends at 7 (since both intervals include 7). Therefore, the interval where both functions are decreasing is (3, 7]. This means that for x-values between 3 (exclusive) and 7 (inclusive), both Function 1 and Function 2 are decreasing.

Representing the Decreasing Interval on a Number Line

Representing the solution on a number line is a crucial step in visualizing the interval where both functions are decreasing. A number line is a visual tool that helps us understand and communicate intervals clearly.

To represent the interval (3, 7] on a number line, we follow these steps:

1. Draw a number line: Draw a horizontal line and mark the relevant numbers, in this case, 3 and 7. You might also include other numbers for context, but 3 and 7 are the most important.
2. Use appropriate endpoints:
   • For an open endpoint (like the 3 in (3, 7]), use an open circle (o). This indicates that the endpoint is not included in the interval.
   • For a closed endpoint (like the 7 in (3, 7]), use a closed circle (•). This indicates that the endpoint is included in the interval.
3. Shade the interval: Shade the region on the number line between the two endpoints. This shaded region represents all the numbers within the interval.

Therefore, on the number line:

• There would be an open circle at 3, indicating that 3 is not included.
• There would be a closed circle at 7, indicating that 7 is included.
• The region between 3 and 7 would be shaded, representing all the numbers between 3 and 7.

This visual representation clearly shows the interval (3, 7], which is the solution to the problem.

Importance of Understanding Decreasing Intervals

Understanding decreasing intervals is a fundamental concept in mathematics, with applications in various fields. Here's why it's so important:

• Calculus: In calculus, identifying decreasing intervals is crucial for determining the local extrema (maxima and minima) of a function. The derivative of a function is negative in decreasing intervals, a key concept in optimization problems.
• Graphing Functions: Knowing where a function is increasing or decreasing helps in sketching its graph accurately. This provides a visual understanding of the function's behavior.
• Optimization: Many real-world problems involve finding the maximum or minimum value of a function (e.g., maximizing profit, minimizing cost). Identifying decreasing intervals helps narrow down the search for optimal solutions.
• Modeling: In mathematical modeling, understanding how a function changes over time or with respect to other variables is essential. Decreasing intervals can represent situations where a quantity is decreasing, such as population decline or the decay of a radioactive substance.

Conclusion

Identifying and representing decreasing intervals is a critical skill in mathematics. We explored how to determine the decreasing intervals of a function algebraically and graphically. We also learned how to find the intersection of intervals and represent the solution on a number line. By understanding these concepts, you'll be well-equipped to tackle more complex problems in calculus and other areas of mathematics. Remember, practice is key to mastering these skills. Work through various examples and problems to solidify your understanding.

This article provides a comprehensive guide to understanding and representing decreasing intervals. By combining algebraic analysis, graphical interpretation, and the use of a number line, we can effectively solve problems involving function behavior. This knowledge is not only essential for academic pursuits but also valuable for real-world applications where understanding trends and optimization is crucial. Continue exploring these concepts, and you'll find your mathematical toolkit growing steadily.