Finding the Range of the Function y (8x - 16) / x^2

Introduction: In this article, we explore the range of the function y (8x - 16) / x^2. Understanding the range is crucial in many applications, whether in mathematics, physics, or engineering. We will delve into the steps required to find the range of this function, including its domain, using calculus and algebraic methods.

Understanding the Function and Its Domain

The given function is y (8x - 16) / x^2. To find the range, we first need to identify the domain of the function, which is the set of all possible values of x for which y is defined. Clearly, y is a real finite number for all real and finite x except when x 0. Therefore, the domain of the function is:

Domain { x ∈ R : x ≠ 0 }

R denotes the set of real numbers. We assume x is finite.

Expressing the Function in a More Tractable Form

To simplify the analysis, let's rewrite the function in a more manageable form:

y 8/x - 16/x^2 - (16/x^2 8/x)

Further simplifying, we get:

y - (16/x 8/x^2) - (4/x - 1)^2 - 1

From this expression, it is clear that the term (4/x - 1)^2 is always non-negative for any real x ≠ 0. Therefore, the maximum value of the term (4/x - 1)^2 is 0, which occurs when 4/x 1, i.e., x 4. Substituting this value back, we get:

y - 0 - 1 -1

Consequently, the value of y is always less than or equal to -1. This means the range of the function is:

Range { y ∈ R : y ≤ -1 }

The equality holds when x 4.

Graphical Representation

The function y (8x - 16) / x^2 exhibits a behavior that diverges to negative infinity as x approaches 0 from both directions. It also has a maximum value of -1 at (4, -1).

A graphical representation of this function would show the curve approaching negative infinity on both sides of 0 and reaching -1 at the point (4, -1).

Conclusion and Additional Resources

Through the above analysis, we have determined that the range of the function y (8x - 16) / x^2 is all real numbers less than or equal to -1. This detailed solution and graph can be explored further through the link provided:

[Detailed Solution and Graph](#)

For a comprehensive understanding of the range of functions, one might also refer to related topics such as the domain of functions, optimization through calculus, and understanding the behavior of rational functions.