Analyzing The Shape And Direction Of V(x) = { -2/7x If X < -1, 7/3 * Cube_root(x) If X >= -1 }

In this comprehensive article, we will delve into the intricate details of the function v(x), defined piecewise as follows:

$v(x)=\left\{\begin{array}{cl}-\frac{2}{7} x & \text { if } x<-1 \\\frac{7}{3} \sqrt[3]{x} & \text { if } x \geq-1\end{array}\right.$

Our primary focus will be on identifying the general shape and direction of the graph of this function, particularly on specific intervals. This involves a thorough analysis of each piece of the function and how they connect at the boundary point, x = -1. We will explore the behavior of the linear part for x < -1 and the cube root part for x ≥ -1. Understanding these characteristics is crucial for sketching the graph and comprehending the overall nature of the function. Let's embark on this mathematical journey to dissect and understand the nuances of v(x).

Analyzing the Function v(x)

Piecewise Definition

The function v(x) is a piecewise function, which means it is defined by different expressions over different intervals of its domain. This particular function is split into two parts:

1. For x < -1, the function is defined as v(x) = -2/7x. This is a linear function.
2. For x ≥ -1, the function is defined as v(x) = (7/3)∛x. This is a cube root function.

The point x = -1 is the critical point where the function transitions from one definition to another. It's essential to examine the behavior of the function around this point to understand its continuity and overall shape.

Interval Discussion

To fully grasp the general shape and direction of v(x), we'll break down the analysis by interval. This approach allows us to focus on the distinct characteristics of each piece of the function.

For x < -1: Linear Behavior

When x < -1, the function v(x) = -2/7x represents a straight line. To understand the general shape and direction, we need to consider two key aspects of a linear function: the slope and the y-intercept (though the y-intercept isn't directly relevant when considering x < -1 due to the domain restriction).

• Slope: The slope of this line is -2/7. A negative slope indicates that the line is decreasing, meaning as x increases, v(x) decreases. In simpler terms, the line slopes downward from left to right. The absolute value of the slope (2/7) tells us the steepness of the line. Since 2/7 is less than 1, the line is not very steep; it's a relatively gentle slope.
• Direction: As x approaches negative infinity, v(x) will approach positive infinity. This is because a negative number multiplied by another negative number yields a positive number. Conversely, as x approaches -1 from the left, v(x) will approach -2/7 * (-1) = 2/7. This gives us an idea of where the linear part of the function ends and how it connects (or doesn't) with the next piece.

In summary, for x < -1, the graph of v(x) is a straight line that slopes downward from left to right, starting at positive infinity and decreasing towards the point (-1, 2/7). Understanding this linear behavior is essential for visualizing the overall shape of the function.

For x ≥ -1: Cube Root Behavior

When x ≥ -1, the function v(x) = (7/3)∛x represents a cube root function. Cube root functions have a distinctive shape that we need to understand to describe the general shape and direction of this piece of v(x).

• General Shape of Cube Root Functions: Cube root functions, in the form of f(x) = a∛x, have an S-like shape. They start from negative infinity as x approaches negative infinity, pass through the origin (0, 0), and extend to positive infinity as x approaches positive infinity. However, our function has a coefficient of 7/3 and is only defined for x ≥ -1, so we need to consider these modifications.
• Coefficient 7/3: The coefficient 7/3 stretches the cube root function vertically. This means that for any given value of x, the value of v(x) will be 7/3 times the value of the basic cube root function ∛x. This vertical stretch doesn't change the fundamental S-shape, but it makes the function increase more rapidly.
• Domain Restriction x ≥ -1: This restriction is crucial. It means we only consider the part of the cube root function that starts at x = -1. At x = -1, v(-1) = (7/3)∛(-1) = -7/3. This tells us that the cube root part of the function starts at the point (-1, -7/3).
• Direction: As x increases from -1, the cube root of x also increases. Since we are multiplying the cube root by a positive number (7/3), v(x) will also increase. As x approaches positive infinity, v(x) will also approach positive infinity. The rate of increase, however, slows down as x gets larger, characteristic of cube root functions.

In summary, for x ≥ -1, the graph of v(x) is a stretched cube root function that starts at the point (-1, -7/3) and increases towards positive infinity. The general shape is an S-like curve that's been vertically stretched, and the direction is upward as x increases.

Connecting the Pieces at x = -1

Now that we've analyzed each piece of the function separately, we need to understand how they connect at the critical point x = -1. This involves comparing the function values and the slopes (or rates of change) as x approaches -1 from both sides.

• Value from the Left (x < -1): As we saw earlier, as x approaches -1 from the left, v(x) approaches 2/7. So, the linear part of the function approaches the point (-1, 2/7).
• Value from the Right (x ≥ -1): At x = -1, the cube root part of the function is v(-1) = -7/3. This means the cube root part starts at the point (-1, -7/3).
• Discontinuity: The function values do not match at x = -1. The linear part approaches 2/7, while the cube root part starts at -7/3. This indicates that the function v(x) is discontinuous at x = -1. There is a jump in the graph at this point.
• Direction and Slope: The linear part has a constant negative slope of -2/7. The cube root part starts with a relatively steep positive slope (due to the vertical stretch) and then the slope decreases as x increases. This change in slope contributes to the overall S-like shape of the cube root portion.

The connection (or lack thereof) at x = -1 is a crucial aspect of the function's general shape. The discontinuity creates a distinct break in the graph, which is an essential feature to note when sketching the function.

Conclusion

In conclusion, the function v(x) exhibits distinct behaviors on the intervals x < -1 and x ≥ -1. For x < -1, it's a decreasing linear function approaching the point (-1, 2/7). For x ≥ -1, it's a vertically stretched cube root function starting at the point (-1, -7/3) and increasing towards positive infinity. The function is discontinuous at x = -1, creating a jump in the graph.

Understanding the general shape and direction of v(x) on these intervals is pivotal for sketching the graph accurately. The linear part provides a straight, downward-sloping segment, while the cube root part contributes an S-like curve that starts below the linear part and rises upwards. The discontinuity at x = -1 further defines the unique characteristics of this piecewise function. This detailed analysis equips us with the necessary insights to visualize and interpret the behavior of v(x) effectively.