Domain Range And Discontinuities Of F(x) G(x) And H(x)

In the realm of mathematics, functions serve as fundamental building blocks for modeling and understanding various phenomena. This article delves into a comprehensive analysis of three distinct types of functions: polynomial, radical, and rational. We will explore their unique characteristics, focusing on their domains, ranges, and interrelationships. Specifically, we will examine the functions f(x) = x⁴ - 4x³ - 2x² + 12x + 9, g(x) = √(x² - 2x - 3), and h(x) = (-x² + 1) / (x² - 2x - 3), dissecting their properties and highlighting their individual behaviors.

Part A: Comparing the Domain and Range of Polynomial Function f(x) and Radical Function g(x)

In this section, we embark on a comparative journey to analyze the domain and range of the polynomial function f(x) = x⁴ - 4x³ - 2x² + 12x + 9 and the radical function g(x) = √(x² - 2x - 3). Understanding the domain and range of a function is crucial as it defines the set of possible input values (x-values) and the corresponding set of output values (y-values). This comparison will shed light on the fundamental differences between these two types of functions.

Delving into the Polynomial Function f(x)

Polynomial functions, characterized by their smooth and continuous nature, hold a prominent position in the mathematical landscape. Our focus function, f(x) = x⁴ - 4x³ - 2x² + 12x + 9, exemplifies this category. Let's dissect its domain and range to grasp its behavior comprehensively.

Unveiling the Domain of f(x)

The domain of a function encompasses all permissible input values (x-values) for which the function yields a real output. For polynomial functions, this domain extends across the entire spectrum of real numbers. This stems from the fact that polynomial expressions involve only addition, subtraction, and non-negative integer exponentiation of the variable x, operations that are well-defined for all real numbers. Therefore, the domain of f(x) = x⁴ - 4x³ - 2x² + 12x + 9 encompasses all real numbers, symbolized as (-∞, ∞). This vast domain signifies that we can input any real number into the function, and it will produce a corresponding real output.

Exploring the Range of f(x)

The range of a function encompasses the set of all possible output values (y-values) that the function can generate. Determining the range of a polynomial function necessitates a deeper understanding of its behavior. Unlike the domain, the range of a polynomial function is not always the set of all real numbers. For f(x) = x⁴ - 4x³ - 2x² + 12x + 9, we observe that it is a polynomial of even degree (degree 4) with a positive leading coefficient (1). This characteristic implies that as x approaches positive or negative infinity, the function f(x) also approaches positive infinity. In simpler terms, the graph of the function rises indefinitely on both ends. To pinpoint the minimum value of the function, we can employ calculus techniques, such as finding critical points by setting the derivative equal to zero and analyzing the second derivative. Alternatively, we can utilize graphing tools or software to visualize the function's behavior. Through these methods, we discover that the minimum value of f(x) is -4, which occurs at x = 1. Therefore, the range of f(x) is [-4, ∞), signifying that the function's output values are always greater than or equal to -4.

Investigating the Radical Function g(x)

Radical functions, distinguished by the presence of a radical sign (√), introduce a different set of constraints on their domains and ranges. Our focus function, g(x) = √(x² - 2x - 3), exemplifies this type of function. Let's delve into its domain and range to unravel its unique behavior.

Deciphering the Domain of g(x)

The domain of a radical function is governed by the expression under the radical sign, known as the radicand. For square root functions, the radicand must be non-negative to ensure a real-valued output. This constraint arises from the fact that the square root of a negative number is not defined within the realm of real numbers. Therefore, for g(x) = √(x² - 2x - 3), we need to identify the values of x for which the radicand, x² - 2x - 3, is greater than or equal to zero. To achieve this, we can solve the inequality x² - 2x - 3 ≥ 0. Factoring the quadratic expression, we get (x - 3)(x + 1) ≥ 0. Analyzing the sign of the factors, we find that the inequality holds true when x ≤ -1 or x ≥ 3. Consequently, the domain of g(x) is (-∞, -1] ∪ [3, ∞), indicating that the function is defined only for x-values less than or equal to -1 or greater than or equal to 3.

Unveiling the Range of g(x)

The range of a radical function is determined by the possible output values that the radical expression can attain. For the square root function g(x) = √(x² - 2x - 3), the output is always non-negative because the square root of a non-negative number is non-negative. Furthermore, as x moves away from the interval [-1, 3], the radicand x² - 2x - 3 increases, leading to larger output values for g(x). The minimum value of the radicand occurs at the endpoints of the domain, x = -1 and x = 3, where the radicand equals zero. Consequently, the minimum value of g(x) is √(0) = 0. As x approaches positive or negative infinity, the radicand also approaches positive infinity, causing g(x) to approach positive infinity as well. Therefore, the range of g(x) is [0, ∞), signifying that the function's output values are always non-negative.

A Comparative Summary

In summary, the polynomial function f(x) boasts an unrestricted domain, encompassing all real numbers, while its range is bounded below by -4. In contrast, the radical function g(x) exhibits a restricted domain, defined by the non-negativity of its radicand, and its range encompasses all non-negative real numbers. These distinctions underscore the fundamental differences between polynomial and radical functions, highlighting their unique behaviors and characteristics.

Part B: Analyzing the Discontinuities of the Rational Function h(x)

Now, let's turn our attention to the rational function h(x) = (-x² + 1) / (x² - 2x - 3). Rational functions, defined as the ratio of two polynomials, introduce the concept of discontinuities, points where the function is not continuous. These discontinuities arise when the denominator of the rational function equals zero, leading to undefined values. Understanding these discontinuities is crucial for a comprehensive analysis of rational functions.

Identifying Potential Discontinuities

To pinpoint the discontinuities of h(x) = (-x² + 1) / (x² - 2x - 3), we need to identify the values of x for which the denominator, x² - 2x - 3, equals zero. Setting the denominator to zero, we obtain the equation x² - 2x - 3 = 0. This quadratic equation can be solved by factoring:

(x - 3)(x + 1) = 0

This factorization reveals two solutions: x = 3 and x = -1. These values are potential points of discontinuity because they make the denominator of the rational function equal to zero.

Classifying the Discontinuities

Having identified the potential discontinuities, we need to classify them as either removable or non-removable. This classification hinges on whether the discontinuity can be