What Is The Range Of The Function Y = \sqrt[3]{x + 8}?
The range of a function is a fundamental concept in mathematics, representing the set of all possible output values that the function can produce. Determining the range is crucial for understanding a function's behavior and its limitations. In this comprehensive exploration, we delve into the intricacies of finding the range of the function y = \sqrt[3]{x + 8}. We will embark on a step-by-step journey, unraveling the mathematical principles involved and arriving at a definitive answer. This detailed explanation will not only provide the solution but also enhance your understanding of function ranges in general.
Understanding the Cube Root Function
Before we dive into the specifics of our function, let's establish a firm grasp of the cube root function. The cube root of a number, denoted by , is the value that, when multiplied by itself three times, equals the original number. Unlike square roots, cube roots can handle both positive and negative numbers, as well as zero. For instance, the cube root of 8 is 2 (because 2 * 2 * 2 = 8), and the cube root of -8 is -2 (because -2 * -2 * -2 = -8). This characteristic of cube roots is crucial to understanding why the function y = \sqrt[3]{x + 8} has a particular range.
In contrast to square root functions, which are restricted to non-negative outputs due to the nature of squaring (a negative number squared becomes positive), cube root functions do not have this limitation. This is because cubing a negative number results in a negative number. This key difference means that the basic cube root function, y = \sqrt[3]{x}, can produce any real number as an output. This foundational knowledge is the first step in determining the range of our slightly more complex function.
Analyzing the Function y = \sqrt[3]{x + 8}
Now, let's turn our attention to the function y = \sqrt[3]{x + 8}. This function is a transformation of the basic cube root function. The addition of 8 inside the cube root, (x + 8), represents a horizontal shift. Specifically, it shifts the graph of the basic cube root function 8 units to the left. This horizontal shift affects the domain of the function, but it does not directly impact the range. The range is determined by the possible output values (y-values) of the function.
The critical aspect to consider is that the cube root function itself, as we established earlier, can produce any real number. Adding 8 to x before taking the cube root does not change this fundamental property. No matter what value we substitute for x, the expression x + 8 will result in some real number, and the cube root of that number will also be a real number. This is because the cube root is defined for all real numbers.
To further clarify, let's consider some examples. If x is a very large positive number, x + 8 will also be a very large positive number, and its cube root will be a large positive number. Similarly, if x is a very large negative number (e.g., -1000), then x + 8 will be a negative number (-992), and the cube root of -992 will be a negative number. This ability to produce both very large positive and very large negative outputs is a key indicator that the range spans all real numbers.
Determining the Range
Given our analysis, we can confidently determine the range of the function y = \sqrt[3]{x + 8}. Since the cube root function can produce any real number, and the addition of 8 inside the cube root does not restrict the possible output values, the range of this function is all real numbers. This means that y can take on any value from negative infinity to positive infinity.
In mathematical notation, we express this range as: (-∞, ∞). This notation signifies that the function's output values can extend indefinitely in both the negative and positive directions. This is a crucial distinction from functions like square root functions, which have a restricted range due to the nature of the square root operation.
Therefore, the correct answer to the question "What is the range of the function y = \sqrt[3]{x + 8}?" is A. -∞ < y < ∞. This comprehensive explanation has detailed the reasoning behind this answer, emphasizing the properties of cube root functions and the impact of transformations on their ranges.
Why Other Options are Incorrect
To solidify our understanding, let's briefly discuss why the other options provided are incorrect. This will reinforce the concept of range and highlight the specific characteristics of our function.
- B. -8 < y < ∞: This option suggests that the range is restricted to values greater than -8. However, as we've established, the cube root function can produce negative outputs. For example, if we input a value of x that makes x + 8 negative, the cube root will also be negative. Therefore, this option is incorrect.
- C. 0 ≤ y < ∞: This option proposes that the range consists of only non-negative values. This is characteristic of square root functions, but not cube root functions. As we've seen, cube roots can be negative, so this option is also incorrect.
- D. 2 ≤ y: This option implies that the range starts at 2 and extends to positive infinity. This is not accurate because the cube root function, even with the horizontal shift, can produce values less than 2. Therefore, this option is incorrect.
The key takeaway is that the cube root function's ability to handle both positive and negative inputs, and consequently produce both positive and negative outputs, is the defining factor in its range being all real numbers. Understanding this distinction is crucial for correctly determining the range of this and similar functions.
Visualizing the Function
A powerful way to understand the range of a function is to visualize its graph. The graph of y = \sqrt[3]{x + 8} is a transformation of the basic cube root function y = \sqrt[3]{x}. The graph of the basic cube root function passes through the origin (0, 0) and extends infinitely in both the positive and negative x and y directions. It has a characteristic "S" shape.
The graph of y = \sqrt[3]{x + 8} is the same "S" shape, but it has been shifted 8 units to the left. This means that the point that was originally at the origin (0, 0) is now at (-8, 0). However, the fundamental shape and the infinite extension in both the vertical (y) directions remain the same. This visual representation clearly demonstrates that the function can take on any y-value, reinforcing the conclusion that the range is all real numbers.
If you were to sketch this graph or use a graphing calculator, you would observe that the curve covers all possible y-values. There are no gaps or restrictions in the vertical direction. This visual confirmation is a valuable tool for understanding and remembering the concept of range.
Domain Considerations
While our focus has been on the range, it's also important to briefly consider the domain of the function y = \sqrt[3]{x + 8}. The domain is the set of all possible input values (x-values) for which the function is defined. For cube root functions, there are no restrictions on the input values. You can take the cube root of any real number, whether it's positive, negative, or zero.
Therefore, the domain of y = \sqrt[3]{x + 8} is also all real numbers, which can be written as (-∞, ∞). This contrasts with functions like square root functions, which have a restricted domain because you cannot take the square root of a negative number and get a real result.
The fact that both the domain and range of y = \sqrt[3]{x + 8} are all real numbers is a key characteristic of cube root functions. Understanding this property is essential for working with these functions in various mathematical contexts.
Applications of Range in Mathematics
The concept of range is not just an abstract mathematical idea; it has significant applications in various areas of mathematics and beyond. Understanding the range of a function allows us to:
- Analyze function behavior: The range tells us the possible output values of a function, which helps us understand its limitations and how it behaves under different conditions.
- Solve equations: When solving equations involving functions, knowing the range can help us determine if a solution is possible. If the desired output value is not within the range, there is no solution.
- Model real-world phenomena: Many real-world situations can be modeled using functions. Understanding the range of these functions helps us interpret the results in a meaningful way. For example, if a function models the height of an object, the range tells us the possible heights the object can reach.
- Determine inverse functions: The range of a function becomes the domain of its inverse function, and vice versa. Understanding the range is crucial for finding and analyzing inverse functions.
In the context of y = \sqrt[3]{x + 8}, the fact that its range is all real numbers has implications for its inverse function. The inverse function will have a domain of all real numbers, which means it is defined for any input value. This is a direct consequence of the original function's range.
Conclusion
In conclusion, the range of the function y = \sqrt[3]{x + 8} is (-∞, ∞), meaning it includes all real numbers. This is because the cube root function can produce any real number as an output, and the addition of 8 inside the cube root does not alter this fundamental property. We arrived at this answer through a detailed analysis of the cube root function, its transformations, and the concept of range itself.
Understanding the range of a function is a crucial skill in mathematics. It provides valuable insights into the function's behavior and its limitations. By mastering this concept, you'll be better equipped to tackle more complex mathematical problems and apply these principles in real-world contexts.
This exploration has not only provided the answer but also aimed to enhance your understanding of function ranges in general. Remember to consider the fundamental properties of the functions involved and how transformations might affect their range. With practice, you'll become proficient in determining the range of various functions and appreciating their unique characteristics.