Comparing Domain And Range Of Functions F(x)=3x^2, G(x)=1/(3x), And H(x)=3x

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Understanding the domain and range of functions is crucial in mathematics. When we consider different functions, comparing their domains and ranges helps us grasp their behavior and characteristics. This article delves into the functions f(x) = 3x², g(x) = 1/(3x), and h(x) = 3x, and will provide a comprehensive comparison of their domains and ranges. By examining these functions, we will determine which statements accurately reflect their properties. This exploration is essential for anyone looking to strengthen their understanding of function analysis and mathematical concepts.

Function Definitions

Before diving into the comparison, let's clearly define the functions we're examining:

  • f(x) = 3x²: This is a quadratic function, where the input x is squared and then multiplied by 3. The graph of this function is a parabola.
  • g(x) = 1/(3x): This is a rational function, where 1 is divided by 3 times x. This function represents a hyperbola.
  • h(x) = 3x: This is a linear function, where the input x is multiplied by 3. The graph is a straight line.

Domains of the Functions

Domain of f(x) = 3x²

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the quadratic function f(x) = 3x², there are no restrictions on the values of x that can be inputted. You can square any real number and multiply it by 3. Therefore, the domain of f(x) is all real numbers. This can be expressed mathematically as: Domain(f) = (-∞, ∞). Understanding this unrestricted input is crucial for analyzing the function's behavior and its graph, which extends infinitely in both positive and negative directions along the x-axis. The absence of any breaks or asymptotes in the domain reflects the continuous nature of the quadratic function. This characteristic is fundamental in many mathematical applications and problem-solving scenarios where f(x) = 3x² might be used. The domain's simplicity also highlights the foundational properties of quadratic functions in mathematical analysis.

Domain of g(x) = 1/(3x)

The domain of the rational function g(x) = 1/(3x) is slightly more complex. A rational function is undefined when the denominator is equal to zero. In this case, the denominator is 3x, which equals zero when x = 0. Therefore, x = 0 must be excluded from the domain. All other real numbers are permissible inputs. Thus, the domain of g(x) consists of all real numbers except 0. This can be expressed in interval notation as: Domain(g) = (-∞, 0) U (0, ∞). This exclusion of zero significantly shapes the graph of g(x), resulting in a vertical asymptote at x = 0. Understanding this domain restriction is vital for accurate graphing and analysis of rational functions. The behavior of g(x) near x = 0 is a key characteristic, showcasing the impact of the domain on the function's overall properties and its applications in various mathematical contexts.

Domain of h(x) = 3x

The domain of the linear function h(x) = 3x is straightforward. Similar to the quadratic function f(x), there are no restrictions on the values of x that can be inputted into h(x). You can multiply any real number by 3, and the result will be a real number. Consequently, the domain of h(x) encompasses all real numbers. This is expressed mathematically as: Domain(h) = (-∞, ∞). The unrestricted domain of h(x) reflects the fundamental nature of linear functions, which are continuous and well-defined across the entire number line. This broad domain is a key characteristic that makes linear functions versatile and widely applicable in mathematical modeling and analysis. The simplicity of the domain also underscores the basic properties of linear relationships, which are foundational in many areas of mathematics and its applications.

Ranges of the Functions

Range of f(x) = 3x²

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) = 3x², the output is always non-negative because squaring any real number results in a non-negative value, and multiplying it by 3 maintains this non-negativity. The minimum value of f(x) occurs when x = 0, where f(0) = 3(0)² = 0. As x moves away from 0 in either direction, f(x) increases without bound. Therefore, the range of f(x) is all non-negative real numbers, which can be expressed as: Range(f) = [0, ∞). Understanding this range is crucial for grasping the behavior of the parabola represented by f(x), which opens upwards with its vertex at the origin. The non-negative output values are a key characteristic of this quadratic function, influencing its applications in various mathematical and real-world contexts. The specific range also highlights the function's symmetry around the y-axis, a fundamental property of even functions.

Range of g(x) = 1/(3x)

The range of the rational function g(x) = 1/(3x) is determined by the possible output values. As x approaches infinity, g(x) approaches 0, but never actually reaches it. Similarly, as x approaches negative infinity, g(x) also approaches 0. At x = 0, the function is undefined, creating a vertical asymptote. For any non-zero value of x, g(x) can take on any non-zero real number. Therefore, the range of g(x) is all real numbers except 0. This can be expressed as: Range(g) = (-∞, 0) U (0, ∞). This range reflects the behavior of the hyperbola represented by g(x), which has horizontal and vertical asymptotes at y = 0 and x = 0, respectively. The exclusion of 0 from the range is a key characteristic, influenced by the domain restriction at x = 0. Understanding this range is essential for analyzing the function's properties and its applications in various mathematical contexts.

Range of h(x) = 3x

The range of the linear function h(x) = 3x is the set of all possible output values. Since x can be any real number, and h(x) is simply 3 times x, the output can also be any real number. As x ranges from negative infinity to positive infinity, h(x) likewise ranges from negative infinity to positive infinity. Therefore, the range of h(x) is all real numbers, which is expressed as: Range(h) = (-∞, ∞). The unrestricted range of h(x) is a characteristic feature of linear functions, reflecting their continuous and consistent behavior across the number line. This broad range makes linear functions versatile and widely used in mathematical modeling, analysis, and various real-world applications. The straightforward nature of the range also underscores the fundamental properties of linear relationships, which are foundational in mathematics and beyond.

Comparing Domain and Range

Now, let's compare the domains and ranges of the three functions:

  • Domains:
    • f(x) = 3x²: All real numbers (-∞, ∞)
    • g(x) = 1/(3x): All real numbers except 0 (-∞, 0) U (0, ∞)
    • h(x) = 3x: All real numbers (-∞, ∞)
  • Ranges:
    • f(x) = 3x²: All non-negative real numbers [0, ∞)
    • g(x) = 1/(3x): All real numbers except 0 (-∞, 0) U (0, ∞)
    • h(x) = 3x: All real numbers (-∞, ∞)

From this comparison, we can make several observations. The functions f(x) and h(x) have domains that include all real numbers, while g(x)'s domain excludes 0. The ranges vary significantly: f(x) has a range of non-negative real numbers, g(x)'s range includes all real numbers except 0, and h(x)'s range includes all real numbers. These differences in domains and ranges are crucial for understanding each function's unique characteristics and behavior.

Evaluating Statements

Based on our analysis, let's evaluate the given statements:

Statement A: All of the functions have a unique range.

To evaluate the statement, “All of the functions have a unique range,” we need to examine the range of each function and determine if they are indeed unique. The range of f(x) = 3x² is [0, ∞), which includes all non-negative real numbers. The range of g(x) = 1/(3x) is (-∞, 0) U (0, ∞), which includes all real numbers except 0. The range of h(x) = 3x is (-∞, ∞), which includes all real numbers. Comparing these ranges, it is clear that each function has a distinct set of output values. f(x)'s range is limited to non-negative numbers, g(x) excludes 0, and h(x) encompasses all real numbers. This variance in the ranges demonstrates that each function behaves differently in terms of its possible output values. Therefore, the statement is accurate because no two functions share the exact same range. This uniqueness is a key aspect of understanding the individual characteristics of these functions and their applications in mathematical contexts. Thus, we can confirm that the assertion regarding the unique ranges of the functions is valid based on our comprehensive analysis.

Statement B

(The content for evaluating statement B is missing in the prompt. To provide a complete answer, the statement needs to be provided.)

Conclusion

In conclusion, analyzing the domain and range of functions is fundamental to understanding their behavior and properties. By comparing the domains and ranges of f(x) = 3x², g(x) = 1/(3x), and h(x) = 3x, we can see how different types of functions exhibit unique characteristics. This exploration not only enhances our mathematical understanding but also provides a basis for solving more complex problems and applications in various fields. The careful examination of these functions' domains and ranges underscores the importance of such analysis in mathematical studies and beyond. A thorough understanding of these concepts allows for a deeper appreciation of the nuances and versatility of mathematical functions.