Piecewise Function Range Determine Values Within Range

Which values are within the range of the piecewise-defined function?

In mathematics, piecewise-defined functions are fascinating entities. These functions are defined by multiple sub-functions, each applying to a specific interval of the domain. Understanding the range of such a function requires a careful examination of each sub-function and its corresponding domain. In this article, we will delve into the process of determining the range of a piecewise-defined function, focusing on the given example.

Defining Piecewise-Defined Functions

A piecewise-defined function is, in essence, a function that is defined by multiple sub-functions, each of which applies to a certain interval of the domain. These sub-functions piece together to create the overall function, hence the name 'piecewise'. Each sub-function has its own unique formula and a specified domain over which it is valid. Piecewise-defined functions are essential tools for modeling real-world scenarios where relationships change abruptly or have different behaviors across different intervals. The most critical aspect of working with these functions is to understand the domain restrictions for each piece, as this dictates which formula applies for a given input value.

Analyzing the Given Piecewise-Defined Function

Let's consider the piecewise-defined function provided:

$f(x)=\left\{\begin{array}{ll}2 x+2, & x< -3 \\ x, & x=-3 \\ -x-2, & x> -3\end{array}\right.$

This function is composed of three sub-functions, each defined over a specific interval of the domain. To fully grasp the behavior of this function, we need to analyze each piece separately and then consider how they fit together.

First Sub-function: 2x + 2 for x < -3

For values of x less than -3, the function is defined as f(x) = 2x + 2. This is a linear function with a slope of 2 and a y-intercept of 2. As x approaches -3 from the left, the function approaches 2(-3) + 2 = -4. However, it's crucial to note that since the inequality is strict (x < -3), the function never actually reaches -4. This creates an open endpoint at the value -4 for this piece of the function.

To determine the range for this sub-function, we need to consider the behavior of the linear equation as x decreases towards negative infinity. As x becomes increasingly negative, 2x + 2 also becomes increasingly negative. Therefore, the range of this sub-function extends from negative infinity up to, but not including, -4. This can be represented as (-∞, -4).

Second Sub-function: x for x = -3

At x = -3, the function is simply defined as f(x) = x. This means that when x is exactly -3, the function value is also -3. This sub-function provides a single point on the graph of the piecewise function, specifically the point (-3, -3). This point is crucial because it fills a potential gap in the range that might exist due to the other sub-functions.

In terms of the range, this sub-function contributes a single, discrete value: -3. This value is essential for accurately determining the overall range of the piecewise function, as it represents a specific output value that the function attains at a particular point in its domain.

Third Sub-function: -x - 2 for x > -3

For values of x greater than -3, the function is defined as f(x) = -x - 2. This is another linear function, but with a negative slope of -1 and a y-intercept of -2. As x approaches -3 from the right, the function approaches -(-3) - 2 = 1. Similar to the first sub-function, since the inequality is strict (x > -3), the function never actually reaches 1. This creates an open endpoint at the value 1 for this piece of the function.

To determine the range for this sub-function, we need to consider its behavior as x increases towards positive infinity. Because the slope is negative, as x becomes increasingly positive, -x - 2 becomes increasingly negative. Therefore, the range of this sub-function extends from negative infinity up to, but not including, 1. This can be represented as (-∞, 1).

Determining the Overall Range

Now that we have analyzed each sub-function individually, we can combine our findings to determine the overall range of the piecewise-defined function. The range is the set of all possible output values (y-values) that the function can produce.

• The first sub-function (2x + 2 for x < -3) contributes the interval (-∞, -4) to the range.
• The second sub-function (x for x = -3) contributes the single value -3 to the range.
• The third sub-function (-x - 2 for x > -3) contributes the interval (-∞, 1) to the range.

To find the overall range, we need to take the union of these intervals and the single value:

Range = (-∞, -4) ∪ {-3} ∪ (-∞, 1)

Since (-∞, -4) and (-∞, 1) are both intervals extending to negative infinity, their union is simply (-∞, 1). We also need to include the single value -3. Therefore, the overall range of the piecewise-defined function is:

Range = (-∞, 1)

Checking the Given Values

Now, let's check which of the given y-values fall within the range (-∞, 1): y = -6, y = -4, y = -3, y = 0, y = 1, y = 3.

• y = -6: This value is less than 1 and therefore within the range.
• y = -4: This value is not within the range because the range extends up to, but does not include, -4.
• y = -3: This value is within the range as it is a specific point included in the range.
• y = 0: This value is less than 1 and therefore within the range.
• y = 1: This value is not within the range because the range extends up to, but does not include, 1.
• y = 3: This value is not within the range because it is greater than 1.

Conclusion

In conclusion, the values that fall within the range of the given piecewise-defined function are -6, -3, and 0. Determining the range of a piecewise function requires a thorough analysis of each sub-function and its domain. By carefully considering the behavior of each piece and combining their contributions, we can accurately identify the set of all possible output values for the function. Understanding the range is crucial for various mathematical applications, including solving equations, graphing functions, and modeling real-world phenomena.

Which of the following values are within the range of the piecewise-defined function?

Piecewise Function Range Determine Values within Range