Understanding the Domain and Range of Functions

In the field of mathematics, specifically with functions, understanding the domain and range is crucial. This article will explore how to identify the domain and range of a given function, focusing on the function fx -3x2. We will also discuss the importance of standard function sets and transformations in answering such questions efficiently.

Introduction to Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) that the function can produce. Understanding these concepts is essential for various applications, including optimization and analysis of mathematical models.

Standard Functions for Analysis

A fundamental set of standard functions is often used to determine the domain and range through elementary transformations. Common examples include basic polynomial functions, trigonometric functions, and exponential functions. By becoming familiar with these standard function sets, one can quickly identify and analyze more complex functions.

Function gx x: Domain and Range Analysis

Let's start by examining the function gx x.

Domain: The domain of gx is all real numbers, denoted as R. This indicates that x can take any real value. Range: The range of gx is [0, ∞). Since g(x) x, the output values are simply the same as the input values, and therefore the range covers all non-negative real numbers.

Transformation of gx to fx

Now, let's consider the function fx 3gx - 2/3. This transformation involves two steps: translating the graph of gx right by 2/3 and then scaling it by a factor of 3. Neither of these transformations affects the domain or the range of the function. Therefore, the domain and range of fx are the same as those of gx.

Domain of fx: Since the domain of gx is all real numbers, the domain of fx is also all real numbers, R. Range of fx: The range of gx is [0, ∞). After scaling by 3, the range becomes [0, ∞), and the rightward translation 2/3 does not affect the range. Thus, the range of fx is [0, ∞).

Function with Absolute Value: fx -3|x|2

Let's now consider a more complex function, fx -3|x|2.

Domain: There is no restriction on multiplying and adding real numbers, and the absolute value of a quantity is always non-negative. Therefore, the domain of fx is all real numbers, R. Range: Since the absolute value of |x| is non-negative, the expression -3|x|2 will always be non-positive. As x approaches infinity or negative infinity, |x| becomes very large, and thus -3|x|2 approaches negative infinity. However, the smallest value it can reach is 0, when |x| 0. Therefore, the range of fx is [0, ∞).

General Analysis: Domain and Range of Polynomial Functions

In general, the domain of a polynomial function with real coefficients is always all real numbers, R, unless there are restrictions imposed by the function itself (such as division by zero).

The range of a polynomial function can be determined by the degree of the polynomial and the leading coefficient. For a quadratic function like fx ax2, where a 0 (in this case, a -3), the parabola opens downwards, and the vertex represents the maximum value of the function. In this case, the range is (-∞, maximum value].

Conclusion

Understanding the domain and range of functions is essential for various mathematical analyses and applications. By using standard function sets and transformations, one can effectively determine these key properties. The examples provided in this article demonstrate how to analyze the domain and range of functions, particularly polynomial and absolute value functions, and how transformations affect these properties.