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## Understanding Piecewise Functions

In the field of mathematics, a piecewise function refers to a function that is defined by multiple sub-functions, each of which applies to a different interval of the function’s domain. These sub-functions might be defined in different ways, such as using a different equation, or different conditions for the domain of the function.

Piecewise functions are commonly used to model real-world phenomena where different rules or relationships apply in different scenarios. For example, a piecewise function could model the cost of shipping packages, with one rule for packages under a certain weight and a different rule for heavier packages.

## Interpreting the Graph of a Piecewise Function

When given the graph of a piecewise function, it’s essential to interpret the graph by visualizing each piece of the function separately. Pay attention to how the different pieces of the function are connected and how the function behaves at the points where the pieces meet.

## Understanding the Graphed Piecewise Function

The graph presented illustrates a piecewise function with three distinct components. The function is defined by different equations for different intervals of the function’s domain.

The first defined interval of the function occurs from x = -3 to x = -1, where the function is represented by a straight line with a slope of 2 and y-intercept of 1. This segment of the function is indicated by a solid line on the graph.

Next, from x = -1 to x = 1, the function is a parabola with its vertex at (0, 2). This segment of the function is represented by a dashed line.

Finally, for x greater than 1, the function is defined by a straight line with a slope of -1 and a y-intercept of 3. This segment is shown by a dotted line on the graph.

## Completing the Description of the Piecewise Function

The task of completing the description of the piecewise function involves specifying the equations that define the function for each of the three intervals.

For the first segment, which ranges from x = -3 to x = -1, the equation of the line can be represented by:

y = 2x + 6

Next, for the parabolic segment that ranges from x = -1 to x = 1, the equation of the parabola can be expressed as:

y = (x – 1)^2 + 2

Lastly, for the third segment that occurs for x greater than 1, the equation of the line is given by:

y = -x + 3

It is important to note that the use of different line styles (solid, dashed, and dotted) for the three segments of the graph is a visual representation of the different parts of the piecewise function.

## Understanding the Behavior of the Function

By analyzing the equations that define each segment of the piecewise function, we can gain insight into the behavior of the function across its domain.

For the first segment, the function exhibits a linear relationship with a positive slope, causing it to increase as x increases within the specified interval.

Moving on to the parabolic segment, the function reaches a minimum value at x = 1 and increases as x moves away from this point in either direction.

Lastly, the third segment depicts a linear relationship with a negative slope, resulting in a decrease in the function’s value as x increases beyond 1.

## Connecting the Pieces of the Function

At the points where the different segments of the function meet, it is essential to consider the behavior of the function to ensure a smooth transition from one segment to the next.

In this particular piecewise function, the points of transition occur at x = -1 and x = 1. It is evident from the graph that at these points, the function is continuous and exhibits a smooth connection between the different segments. This implies that the function has no discontinuities at these points, and the transition between segments is seamless.

## Identifying Domain and Range

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In the case of the piecewise function, the domain is determined by the collection of all x-values for which each segment of the function is defined.

For the given piecewise function, the domain consists of all real numbers, as each segment of the function is defined across the entire real number line. Therefore, the domain is represented by the interval (-∞, ∞).

On the other hand, the range of a function refers to the set of all possible output values (y-values) that the function can produce for its domain. By observing the graph of the piecewise function, we can determine that the range spans from negative infinity to positive infinity, encompassing all real numbers.

## Identifying Key Features of the Function

When describing a piecewise function, it’s important to identify and articulate the key features of the function. This includes characteristics such as the intercepts, maxima, minima, and any points of inflection.

For the given piecewise function, the key features can be summarized as follows:

– Y-intercept: The function intersects the y-axis at y = 6, where x = -3.

– Minimum Point: The parabolic segment of the function reaches a minimum value of y = 2 at x = 1.

– X-intercept: The function intersects the x-axis at x = 1, where y = 0.

It is also worth noting that the function does not possess any points of inflection, as the behavior of the function remains consistent across each segment.

## Applications of Piecewise Functions

Piecewise functions find widespread application in various fields, particularly in the fields of mathematics, economics, physics, engineering, and computer science. They are commonly used to model real-world scenarios where different rules or relationships apply in different situations.

In economics, piecewise functions could be utilized to model cost or revenue functions that change based on different intervals of production or sales. In engineering, piecewise functions could be employed to describe the behavior of mechanical systems under different operating conditions.

Furthermore, in computer science, piecewise functions are valuable for creating algorithms that behave differently based on certain conditions or inputs.

## Conclusion

In conclusion, the description of the piecewise function graphed above has been completed by identifying the equations that define its segments, interpreting its behavior, identifying its domain and range, and highlighting its key features. Understanding and effectively describing piecewise functions is crucial for their application in various mathematical and real-world contexts. These functions provide a valuable tool for modeling complex relationships and behaviors that change across different intervals, making them an essential concept in the study of mathematics and its applications.