Domain And Range Of Quadratic Function Y=(x+3)^2-5

In mathematics, understanding the domain and range of a function is crucial for analyzing its behavior and properties. For the given quadratic function $y = (x+3)^2 - 5$, we aim to determine its domain and range accurately. This article provides a comprehensive exploration of the concepts of domain and range, and how they apply to quadratic functions, culminating in the correct answer and a thorough explanation.

Domain and Range: Fundamental Concepts

Before diving into the specifics of the given function, let's clarify the core concepts of 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 simpler terms, it's the set of all x-values that you can plug into the function without causing any mathematical errors, such as division by zero or taking the square root of a negative number. The range, on the other hand, is the set of all possible output values (y-values) that the function can produce. It represents the spread of y-values that result from plugging in all the valid x-values from the domain. To put it another way, the range represents the set of all possible outcomes of the function. Understanding these concepts is pivotal for effectively analyzing functions and their graphical representations. For example, if a function represents a real-world scenario, the domain might represent the set of all possible times, and the range might represent the set of all possible temperatures. In the context of graphs, the domain corresponds to the extent of the function along the x-axis, while the range corresponds to the extent along the y-axis. Different types of functions have different characteristic domains and ranges. Linear functions, for example, typically have a domain and range of all real numbers, while rational functions may have restricted domains due to potential division by zero. Likewise, square root functions have restricted domains since the input to the square root cannot be negative. When analyzing a function, it's essential to consider its mathematical structure to determine its domain and range. This involves identifying any operations that might impose restrictions on the input or output values. With a solid grasp of domain and range, one can better understand the behavior of a function and its applications across various mathematical and real-world contexts. This understanding is not only crucial for theoretical mathematics but also for practical applications, such as modeling physical systems, analyzing data, and solving engineering problems.

Analyzing the Domain of y=(x+3)^2-5

When determining the domain of the quadratic function $y = (x+3)^2 - 5$, it is essential to consider any restrictions on the input variable, x. The domain represents the set of all possible x-values for which the function produces a valid output. In the case of quadratic functions, such as the one provided, there are no inherent restrictions on the x-values. This is because you can square any real number, multiply it by a constant, or add/subtract constants without encountering any mathematical errors. For example, we do not have to worry about dividing by zero, which would exclude certain values from the domain, nor do we need to avoid taking the square root of a negative number, which would impose another restriction. Therefore, the domain of a quadratic function, in general, and specifically for $y = (x+3)^2 - 5$, encompasses all real numbers. This means that you can input any real number for x, and the function will yield a real number as an output. Graphically, this implies that the parabola representing the quadratic function extends infinitely in both the left and right directions along the x-axis. To illustrate this point, consider a few examples. If we substitute x with a large positive number, such as 1000, the function will produce a real number. Similarly, if we substitute x with a large negative number, such as -1000, the function will also yield a real number. Even if we substitute x with zero or any fractional value, the function will still produce a valid output. Since there are no values of x that would cause the function to be undefined, the domain is all real numbers. This can be expressed in interval notation as $(-\infty, \infty)$, which signifies that the domain includes all numbers from negative infinity to positive infinity. In conclusion, the absence of any restrictions on x in the quadratic function $y = (x+3)^2 - 5$ confirms that its domain is indeed all real numbers, making it a fundamental characteristic of this type of function. This understanding is critical for further analysis of the function's behavior and properties.

Determining the Range of y=(x+3)^2-5

To determine the range of the quadratic function $y = (x+3)^2 - 5$, we need to identify the set of all possible output values (y-values) that the function can produce. Unlike the domain, which we established as all real numbers, the range of a quadratic function is constrained by its parabolic shape. Specifically, the range is affected by the vertex of the parabola and the direction in which the parabola opens. In the given function, $y = (x+3)^2 - 5$, we can recognize that it is in vertex form, which is $y = a(x-h)^2 + k$, where (h, k) represents the vertex of the parabola. In this case, the vertex is at (-3, -5). The coefficient a determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0). Here, a is 1, which is positive, so the parabola opens upwards. This means that the vertex represents the minimum point of the function. Since the parabola opens upwards from the vertex (-3, -5), the minimum y-value of the function is -5. As x moves away from -3 in either direction, the term $(x+3)^2$ will always be non-negative (since squaring any real number results in a non-negative value). Consequently, the y-values will increase above -5. Therefore, the range of the function includes all y-values greater than or equal to -5. In interval notation, this is represented as $[-5, \infty)$. This notation indicates that the range includes -5 and extends to positive infinity. To further clarify, consider that the square of any real number is non-negative. The smallest value $(x+3)^2$ can take is 0, which occurs when $x = -3$. At this point, $y = (0) - 5 = -5$. For any other value of x, $(x+3)^2$ will be greater than 0, making y greater than -5. Hence, the function's output values are always -5 or greater. In summary, by analyzing the vertex form of the quadratic function and considering the direction of the parabola's opening, we can confidently conclude that the range of $y = (x+3)^2 - 5$ is $[-5, \infty)$. This understanding is vital for interpreting the function's behavior and its graphical representation.

Correct Answer and Explanation

After analyzing the domain and range of the function $y = (x+3)^2 - 5$, we can now identify the correct answer. We determined that the domain of the function is all real numbers, which is represented as $(-\infty, \infty)$. The range of the function, on the other hand, is all real numbers greater than or equal to -5, represented as $[-5, \infty)$.

Given the options:

A. Domain: $[-5, \infty)$

Range: $(-\infty, \infty)$

B. Domain: $(-\infty, \infty)$

Range: $[-5, \infty)$

Comparing our findings with the provided options, we can see that:

Option A incorrectly states the domain as $[-5, \infty)$ and the range as $(-\infty, \infty)$. These are not consistent with our analysis.

Option B correctly identifies the domain as $(-\infty, \infty)$ and the range as $[-5, \infty)$. This matches our detailed analysis of the function.

Therefore, the correct answer is B. The domain of $y = (x+3)^2 - 5$ is $(-\infty, \infty)$, and its range is $[-5, \infty)$.

This comprehensive explanation clarifies why option B is the correct choice. The domain is unrestricted, allowing any real number as an input, while the range is bounded below by the vertex of the parabola at y = -5, reflecting the function's minimum output value. Understanding these characteristics is essential for accurately interpreting the behavior and properties of quadratic functions.

Conclusion

In conclusion, determining the domain and range of a function is a fundamental aspect of mathematical analysis. For the quadratic function $y = (x+3)^2 - 5$, the domain is all real numbers, represented as $(-\infty, \infty)$, while the range is $[-5, \infty)$, indicating all real numbers greater than or equal to -5. Understanding these concepts not only provides a comprehensive view of the function's behavior but also lays the groundwork for more advanced mathematical explorations. The correct answer, as identified, is option B, which accurately reflects the function's domain and range.

By meticulously analyzing the function's structure, particularly its vertex form, we were able to identify the vertex at (-3, -5) and determine that the parabola opens upwards. This confirmed that the minimum y-value is -5, thereby defining the lower bound of the range. The absence of any restrictions on the input variable x established the domain as all real numbers. This exercise underscores the importance of understanding the properties of different types of functions, such as quadratic functions, to effectively analyze their domain and range. The domain and range are not merely abstract mathematical concepts; they provide valuable insights into the function's behavior and its applicability in various real-world scenarios. For instance, in physics, the domain might represent time, and the range might represent distance or height. In economics, the domain could represent the quantity of goods produced, and the range could represent the profit. Therefore, a solid grasp of domain and range is crucial for both theoretical mathematics and practical applications. This exploration of the domain and range of the quadratic function $y = (x+3)^2 - 5$ serves as a valuable example of how mathematical analysis can lead to a deeper understanding of functional behavior and its implications.

By mastering these fundamental concepts, students and practitioners alike can confidently tackle more complex mathematical problems and appreciate the elegance and utility of mathematics in various fields.