rational function

简明释义

有理函数

英英释义

例句

1.A common example of a rational function 有理函数 is f(x) = (2x + 3)/(x - 1).

一个常见的 rational function 有理函数 示例是 f(x) = (2x + 3)/(x - 1)。

2.You can find the domain of a rational function 有理函数 by identifying values that make the denominator equal to zero.

通过识别使分母等于零的值，你可以找到 rational function 有理函数 的定义域。

3.The graph of a rational function 有理函数 can have vertical asymptotes where the denominator is zero.

一个 rational function 有理函数 的图形可以在分母为零的地方有垂直渐近线。

4.To analyze the behavior of a rational function 有理函数, you can perform polynomial long division.

为了分析一个 rational function 有理函数 的行为，你可以进行多项式长除法。

5.In calculus, a rational function 有理函数 is defined as the ratio of two polynomial functions.

在微积分中，rational function 有理函数 被定义为两个多项式函数的比率。

作文

In the field of mathematics, a rational function is defined as a function that can be expressed as the quotient of two polynomial functions. This means that if we have two polynomials, say P(x) and Q(x), then a rational function can be written in the form R(x) = P(x)/Q(x), where Q(x) is not equal to zero. The study of rational functions is crucial because it allows mathematicians to understand the behavior of various types of equations and their graphs. Rational functions exhibit unique characteristics that differentiate them from other types of functions. For instance, they can have vertical asymptotes, horizontal asymptotes, and holes in their graphs. Vertical asymptotes occur when the denominator Q(x) approaches zero while the numerator P(x) remains non-zero. This leads to the function approaching infinity or negative infinity, which creates a vertical line on the graph that the function cannot cross. On the other hand, horizontal asymptotes describe the behavior of the rational function as the input values approach positive or negative infinity. These asymptotes help in predicting the end behavior of the function. Additionally, holes in the graph of a rational function occur when both the numerator and denominator share a common factor. When this common factor is canceled out, it results in a point on the graph that is undefined, creating a hole. Understanding these features is essential for graphing rational functions accurately and analyzing their properties. One practical application of rational functions is in economics, where they are used to model cost and revenue functions. For example, if a company has fixed costs and variable costs that depend on the number of products sold, the total cost function can often be represented as a rational function. By analyzing this function, businesses can determine optimal pricing strategies and maximize profits. In addition to economics, rational functions are also used in physics, engineering, and many other fields. They help in solving real-world problems by providing models that can predict outcomes based on varying inputs. The versatility of rational functions makes them a fundamental concept in mathematics, and their applications extend far beyond theoretical studies. To conclude, a rational function is a vital component in the study of mathematics, with significant implications in various fields. By understanding its definition, characteristics, and applications, one can appreciate the importance of rational functions in both academic and practical contexts. Mastering this concept not only enhances mathematical skills but also equips individuals with tools to tackle complex problems in everyday life.

在数学领域，有理函数被定义为可以表示为两个多项式函数的商。这意味着，如果我们有两个多项式，假设为P(x)和Q(x)，那么有理函数可以写成R(x) = P(x)/Q(x)，其中Q(x)不等于零。研究有理函数至关重要，因为它使数学家能够理解各种类型方程及其图形的行为。有理函数展现出独特的特征，使其与其他类型的函数区分开来。例如，它们可以在图形中具有垂直渐近线、水平渐近线和空洞。垂直渐近线发生在分母Q(x)接近零而分子P(x)保持非零时。这导致函数接近正无穷或负无穷，从而在图形上形成一条函数无法穿越的垂直线。另一方面，水平渐近线描述了当输入值趋近于正无穷或负无穷时，有理函数的行为。这些渐近线有助于预测函数的末端行为。此外，有理函数图形中的空洞发生在分子和分母共享一个共同因子时。当这个共同因子被消去时，会导致图形上一个未定义的点，从而形成一个空洞。理解这些特征对于准确绘制有理函数的图形和分析其属性至关重要。有理函数的一个实际应用是在经济学中，它们用于建模成本和收入函数。例如，如果一家公司有固定成本和依赖于销售产品数量的可变成本，那么总成本函数通常可以表示为有理函数。通过分析这个函数，企业可以确定最佳定价策略并最大化利润。除了经济学，有理函数还广泛应用于物理学、工程学以及许多其他领域。它们通过提供模型来帮助解决现实世界的问题，这些模型可以根据变化的输入预测结果。有理函数的多功能性使其成为数学中的基本概念，其应用远远超出了理论研究。总之，有理函数是数学研究中的一个重要组成部分，在各个领域具有重要意义。通过理解其定义、特征和应用，人们可以欣赏到有理函数在学术和实践背景中的重要性。掌握这一概念不仅增强了数学技能，还为个人提供了解决日常生活中复杂问题的工具。