Understanding the Range of y in y -x

Introduction

The equation y -x is a simple linear function representing a straight line with a slope of -1. It is important to understand the properties of such a function, including its domain and range. In this article, we will explore the range of y for the given function, along with the broader concepts of domain and range.

What is the Function y -x?

The function y -x can be understood as a mapping of every value of x to its negative counterpart, y. This means that whatever value of x is input, the output y will be the exact opposite of that value. For example, if x 2, then y -2. If x -3, then y 3.

Domain and Range

Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of y -x, there are no restrictions on the values of x. Thus, x can take any real number value from negative infinity to positive infinity.

Domain of y -x: -∞ to ∞

Range

The range of a function is the set of all possible output values (y-values) that the function can produce. For y -x, since y is simply the numeric opposite of x, and x can be any real number, y can also be any real number. Therefore, y can take any value from negative infinity to positive infinity.

Range of y -x: -∞ to ∞

Graphical Representation

To better understand the function y -x, it's useful to visualize it on a Cartesian plane. A point (x, y) lies on the graph of the function if and only if y -x. This results in a straight line that passes through the origin with a slope of -1. This line is symmetric about the origin and makes a 45-degree angle with the x-axis, crossing the negative quadrant on one side and the positive quadrant on the other.

Properties and Significance

The function y -x has several interesting properties:

Linearity: It is a linear function, meaning its graph is a straight line. Origin Symmetry: The function is symmetric about the origin, meaning any point (x, y) on the line has a corresponding point (-x, -y). Zero Crossings: The function crosses the x-axis at (0, 0) and the y-axis at (0, 0), meaning both the x and y intercepts are zero.

Real-World Applications

Understanding the range of y in the function y -x is useful in various real-world scenarios. For example, in economics, it can model situations where the cost is directly proportional to and opposite in direction to the quantity produced. In physics, it can represent scenarios where one quantity is exactly the opposite of another, such as reactions where the output is the exact opposite of the input.

Conclusion

In summary, the range of y for the function y -x is all real numbers from negative infinity to positive infinity. This concept is fundamental to understanding the behavior of linear functions, their graphical representations, and their applications in a variety of fields.