Range Of Composite Functions If R(x) = 2 - X^2 And W(x) = X - 2
This article dives deep into the process of determining the range of a composite function, using the specific example where and . We aim to find the range of , which means we need to understand how the functions and interact and how their combined effect shapes the possible output values. Mastering composite functions is crucial for various mathematical applications, including calculus and advanced algebra. This comprehensive guide will walk you through each step, ensuring a clear understanding of the underlying concepts and techniques.
1. Deconstructing the Functions: A Foundation for Understanding
Before we tackle the composite function, let's first dissect the individual functions and . This foundational step is critical for comprehending how they interact within the composition. Understanding the behavior of each function independently allows us to predict the behavior of their composition. We need to identify their individual characteristics, such as their domain and range, and any specific properties that might influence the final result. This initial analysis is the cornerstone of solving the problem effectively.
1.1. Analyzing the Function
Letβs delve into the details of . This is a quadratic function, specifically a parabola. Quadratic functions are characterized by their U-shaped graph, which opens either upwards or downwards depending on the coefficient of the term. In this case, the coefficient is -1, which is negative. This tells us that the parabola opens downwards. This downward-opening nature is crucial because it dictates that the function will have a maximum value. The vertex of the parabola will represent this maximum point, and the range will extend from negative infinity up to this maximum value. Understanding this fundamental property of the function is key to determining its range and, consequently, the range of the composite function.
To pinpoint the maximum value, we need to identify the vertex of the parabola. For a quadratic function in the form , the x-coordinate of the vertex is given by . In our case, , so and . Therefore, the x-coordinate of the vertex is . Now, to find the y-coordinate (which represents the maximum value of the function), we substitute back into the function: . Thus, the vertex of the parabola is at the point (0, 2). This tells us that the maximum value of the function is 2.
Since the parabola opens downwards and its maximum value is 2, the range of is all real numbers less than or equal to 2. We can express this mathematically as . This means that the output of can be any number from negative infinity up to and including 2. This understanding of the range of is crucial because it becomes the domain of the outer function in the composite function, which will directly impact the final range of . Therefore, a thorough analysis of is indispensable for solving the problem.
1.2. Analyzing the Function
Next, we turn our attention to the function . This is a linear function. Linear functions are characterized by their straight-line graphs. They have a constant rate of change (slope) and extend infinitely in both directions. In this particular case, represents a simple linear transformation: a vertical shift. Specifically, it shifts the input value down by 2 units. Understanding this shift is crucial to grasping how it will affect the overall range when composed with another function.
The domain of is all real numbers, as there are no restrictions on the values that can be input into the function. For any real number , we can simply subtract 2. Similarly, the range of is also all real numbers. For any real number , we can find an such that (specifically, ). This means that the output of can be any real number. However, this range of all real numbers for considered in isolation does not directly translate to the range of the composite function . The range of will act as the domain for in the composition, thus significantly influencing the final range.
In the context of the composite function, the range of , which we determined to be , becomes the effective domain for . We are not considering all possible inputs for , but only those that result from the output of . This restriction is key to understanding why the range of will not be all real numbers, even though the range of alone is all real numbers. Therefore, analyzing in the context of its composition with is essential for accurately determining the final range.
2. Constructing the Composite Function
Now that we have a firm grasp of the individual functions, we can move on to constructing the composite function . Composite functions are formed by applying one function to the result of another. The notation indicates that we first apply the function to , and then we apply the function to the result. In other words, we're plugging the entire function into the input of the function . This process of function composition creates a new function with its own unique properties and range.
To find the expression for , we substitute into . We know that and . So, to find , we replace the in with the entire expression for :
Simplifying this expression, we get:
Thus, the composite function is equal to . This simplified form is crucial because it allows us to analyze the behavior and range of the composite function more easily. We can now see that is also a quadratic function, but with a significant difference: it's a downward-opening parabola with its vertex at the origin (0, 0). This understanding of the simplified form is the key to determining the range of the composite function.
The process of constructing the composite function highlights the importance of understanding function notation and substitution. The ability to correctly substitute one function into another is a fundamental skill in mathematics, particularly in calculus and advanced algebra. In this case, by correctly substituting into , we transformed a seemingly complex composition into a more manageable expression, . This simplified form makes the subsequent step of determining the range significantly easier.
3. Determining the Range of
Now that we have the simplified form of the composite function, , we can determine its range. The range of a function is the set of all possible output values. To find the range, we need to consider how the function transforms its inputs and what values it can potentially produce.
The function is a parabola that opens downwards. This is because the coefficient of the term is negative (-1). Downward-opening parabolas have a maximum value at their vertex, and their range extends from negative infinity up to this maximum value. This is a crucial characteristic that helps us define the boundaries of the range. The fact that it opens downwards immediately tells us that there will be an upper limit to the output values, and no lower limit other than negative infinity.
To find the vertex of the parabola, we can use the same approach as before. For a quadratic function in the form , the x-coordinate of the vertex is given by . In our case, , so and . Therefore, the x-coordinate of the vertex is . To find the y-coordinate (the maximum value), we substitute back into the function: . Thus, the vertex of the parabola is at the point (0, 0). This tells us that the maximum value of the function is 0.
Since the parabola opens downwards and its maximum value is 0, the range of is all real numbers less than or equal to 0. In interval notation, this is expressed as . This means that the output of the composite function can be any number from negative infinity up to and including 0. No output value will be greater than 0 because of the downward-opening nature of the parabola. This understanding of the range is the final solution to the problem.
The process of determining the range of highlights the importance of recognizing function types and their properties. By identifying the function as a downward-opening parabola, we could immediately deduce the general form of the range. The specific maximum value, determined by the vertex, then pinpointed the exact upper bound of the range. This approach demonstrates a powerful method for analyzing functions and their behavior.
4. Conclusion: Mastering Composite Function Ranges
In conclusion, the range of the composite function is . We arrived at this answer by systematically analyzing the individual functions, constructing the composite function, and then determining the range of the resulting expression. This process involved several key steps: understanding the properties of quadratic and linear functions, correctly applying function composition, and identifying the vertex of the resulting parabola. Mastering these steps is essential for tackling more complex problems involving composite functions.
The key takeaways from this analysis are:
- Understanding the range of the inner function is crucial because it becomes the domain of the outer function in a composite function.
- Recognizing the type of function (linear, quadratic, etc.) helps determine its general behavior and potential range.
- Constructing the composite function by correct substitution is a fundamental skill.
- The vertex of a parabola plays a crucial role in determining its range.
By applying these principles, you can confidently approach problems involving composite functions and determine their ranges effectively. The ability to work with composite functions is a fundamental skill in mathematics, with applications extending to calculus, differential equations, and various other fields. This detailed guide has provided a clear pathway to understanding and mastering this important concept.
Therefore, the correct answer is A. .