Exploring the Domain and Range of the Function f(x) -7x^2 √3x 5

The mathematical analysis of a function involves determining its domain and range. Here, we delve into the function f(x) -7x^2 √3x 5 to understand both of these aspects.

Understanding the Domain

The domain of a function is the set of all input values (x-values) for which the function is defined. In the context of f(x) -7x^2 √3x 5, it's essential to recognize that this function does not contain any denominators, square roots of negative numbers, or other restrictions that would limit the domain. Therefore, this function is defined for all real numbers. Mathematically, we can express the domain as:

Df R,
where R represents the set of all real numbers.

Exploring the Range

The range of a function is the set of all possible output values (y-values) that the function can produce. For the quadratic function f(x) -7x^2 √3x 5, we can follow a systematic approach to determine its range by considering the properties of quadratic equations.

Step 1: Express the Function in Standard Form

First, we can rewrite the function in a standard form to better understand its behavior. The original function is:

y -7x^2 √3x 5

We can rearrange this to form a quadratic equation in terms of x:

7x^2 - √3x - (y - 5) 0

Step 2: Analyze the Quadratic Equation

Since this is a quadratic equation in the form of ax^2 bx c 0, we can use the discriminant to determine the nature of the roots and thus the range of the function. Here, a 7, b -√3, and c -(y - 5). The discriminant, Δ, is given by:

Δ b^2 - 4ac (-√3)^2 - 4(7)(-(y - 5)) 3 28(y - 5) 3 28y - 140 28y - 137

The quadratic equation has real solutions (i.e., the function has real values) if and only if the discriminant is non-negative:

28y - 137 ≥ 0
28y ≥ 137
y ≥ 137/28

This means that the function can produce all real outputs greater than or equal to 137/28. Therefore, the range of the function is:

Rf [137/28, ∞)

Conclusion

To summarize, we have determined that the domain of the function f(x) -7x^2 √3x 5 is the set of all real numbers, ?. The range of the function is all real numbers greater than or equal to 137/28, which is denoted as [137/28, ∞).

Further Exploration

For a deeper understanding, one might explore the graphical representation of the function, analyze its vertex (maximum or minimum point), and study its behavior at the boundaries of its range. These insights can provide a comprehensive view of the function's properties and behavior.

Understanding the domain and range of functions is crucial in many areas of mathematics, including calculus, algebra, and real analysis. By mastering these concepts, students and professionals can better comprehend and apply mathematical principles in various fields.