Exploring the Domain and Range of Quadratic Functions: A Step-by-Step Guide

In this article, we will delve into the concepts of domain and range for quadratic functions, specifically the function y -x - 22 2. We will break down the process of determining these fundamental properties, ensuring a comprehensive and understandable explanation.

Understanding the Basics of Domain and Range

Before we begin, it is crucial to establish a clear definition of what domain and range mean:

Domain: The set of all possible input values (x-values) that a function can accept. In other words, the domain includes all the values of x for which the function is defined. Range: The set of all possible output values (y-values) produced by the function. This encompasses all the values that the function can produce for the given domain.

For many functions, the domain and range can be determined through logical reasoning and mathematical analysis. Let's explore how this is done for the given quadratic function.

Analysis of the Function Y -x - 22 2

The given function is y -x - 22 2. Let's first simplify and rewrite this function to make it easier to understand.

First, let's simplify the expression:

y -x - 4 2 -x - 2

Domain

For this quadratic function, we need to determine the domain. The function involves a linear term (-x) and a constant term (-2). There are no restrictions on x that would make the function undefined (no division by zero and no taking the square root of a negative number). Therefore, the domain is all real numbers:

Domain: (-∞, ∞)

Range

To determine the range, we need to understand the behavior of the function. Since the function is linear and not quadratic (there is no x2 term), the range will not be limited by a parabola's maximum or minimum values. Let's break this down:

Linear Terms: The function is linear and its slope is -1, which means it will decrease as x increases. There are no upper or lower bounds for this linear function. Constant Term: Subtracting 2 from the linear term means the function's values will be offset by -2.

Given that there are no restrictions on x, the function can take on any real value. Therefore, the range of the function is all real numbers:

Range: (-∞, ∞)

However, it is important to note that the function y -x - 2 is a straight line, and the range is all real numbers. For the sake of completeness, let's consider the given quadratic form y -x - 22 2 before simplification:

Vertex Form: The vertex form of a quadratic function is y a(x - h)2 k. Here, we have y -x - 4 2 -x - 2, which simplifies to y -x - 2. Direction of the Parabola: Since the coefficient of x2 is negative (-1), the parabola opens downwards. Vertex: The vertex of the parabola is (2, 2) when the function is written in the form y -(x - 2)2 2.

Considering the vertex (2, 2) and the fact that the parabola opens downwards, the maximum value of y is 2, and as x moves away from 2, y decreases without bound. Therefore, the range is:

Range: (-∞, 2]

Summary

In summary, for the quadratic function y -x - 22 2, we have determined:

Domain: (-∞, ∞) Range: (-∞, 2]

The function is linear and can take on any real value for x, making the range all real numbers. Considering the simplified form, the range is actually all real numbers minus the maximum value at the vertex, which is 2.

Conclusion

Understanding the domain and range of functions is fundamental in mathematics, especially when dealing with quadratic functions. By evaluating the function's behavior and its graphical representation, we can effectively determine its domain and range. This knowledge is crucial for various applications in algebra, calculus, and real-world scenarios.