Exploring the Range of a Function: An In-Depth Analysis

Understanding the range of a function is a fundamental concept in calculus and mathematics. In this article, we delve into the process of finding the range of the function f(x) 1 frac{1}{x^2} - x. We will explore the steps involved in determining the range and highlight key mathematical principles.

Introduction to the Function

The function in question is f(x) 1 frac{1}{x^2} - x. This expression involves a rational term and a polynomial term. To find its range, we must first understand the behavior of these individual components and their interaction.

Step-by-Step Solution

1. f(x) Domain

The function f(x) 1 frac{1}{x^2} - x is a polynomial with a rational term. The domain of the function is all real numbers except where the denominator is zero. Since x^2 is present in the denominator, x ≠ 0.

(Domain: x in (-infty, 0) cup (0, infty))

Breaking Down the Function

2. Expressing f(x) as a Simplified Form

Let's rewrite the function as:

f(x) 1 frac{1 - x^2}{x^2}

Here, we separate the numerator into 1 - x^2. Now, we have:

f(x) 1 frac{1 - x^2}{x^2} 1 frac{1}{x^2} - 1 1 frac{1}{x^2} - x)

Analyzing the Denominator

The term 1 frac{1}{x^2} - x involves a rational part 1 frac{1}{x^2}. The minimum value of the denominator 1 frac{1}{x^2} can be determined using the concept from calculus, specifically the vertex of a parabola.

The minimum value of 1 frac{1}{x^2} occurs at x sqrt{1}), which is 0.5 when x^2 frac{1}{4}) or x pm frac{1}{2}).

(text{Minimum value of } 1 frac{1}{x^2} 1 frac{1}{(frac{1}{2})^2} 1 4 4 - 1 1 frac{3}{4} frac{3}{4})

Behavior at Infinity

The term x tends to infinity as x approaches infinity and negative infinity. As x gets very large (positive or negative), the polynomial term -x will dominate the function, making f(x) approach negative infinity.

Determination of Range

With the minimum value of the denominator as frac{3}{4}), we can determine the range of the function. The maximum value is found when the denominator is minimized, and the value of x 1) gives the maximum value of the polynomial term.

At x 1):
f(1) 1 frac{1}{1^2} - 1 1 1 - 1 1 1 frac{4}{3})

Therefore, the minimum value of the function is 1, and the maximum value is frac{4}{3}) or approximately 1.33.

Conclusion

In conclusion, the range of the function f(x) 1 frac{1}{x^2} - x is (1, frac{4}{3})). The minimum value of the function is 1, and the maximum value is frac{4}{3}), excluding the endpoints due to the nature of the rational term.

Key Takeaways

The range of the function f(x) 1 frac{1}{x^2} - x) is (1, frac{4}{3})). The minimum value of the denominator 1 frac{1}{x^2}) is frac{3}{4}). The maximum value of the function is obtained when the denominator is minimized.

Related Keywords

For those interested in further exploring related mathematical concepts, consider these keywords:

function range calculus minimum value

By understanding these concepts and applying the steps outlined in this article, you can find the range of more complex functions.