### 商科代写|商业数学代写business mathematics代考|CONCEPT OF A FUNCTION

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 商科代写|商业数学代写business mathematics代考|CONCEPT OF A FUNCTION

When mathematics is used to model real-world conditions, it must account for relationships between quantities. For example, the growth of bacteria over time, the amount of sales that follow from a given advertising budget, or the increase in the amount earned when a sum of money is invested in a bond.
As an example, Table $3.1$ illustrates a relationship between the number of cars sold over time, from 2016 to 2021 by Village Distributors, a small new car dealer. Each year’s sales are arranged under the corresponding year, which clearly shows the relationship between the two quantities, year and number of cars’ sold. Now consider Report 3.1, which is both wordier than Table $3.1$ and less useful. The reason is that a clear and direct assignment between individual years and the number of cars sold during each year is not immediately evident.

The notion of two distinct sets of quantities (like years and number of cars sold) and a rule of assignment between the sets, as presented in Table $3.1$ by arranging corresponding entries under each other, is central to the concept of a function. In fact, it describes a function.

Definition 3.1 A function is an assignment rule between two sets, which assigns to each element in the first set exactly one element (but not necessarily a different one) in the second set.

A function therefore has three components: (1) a first set (perhaps years),
(2) a second set (perhaps numbers), and (3) an assignment rule between the

two sets. This rule must be complete in that an assignment must be made to each and every element of the first set. As an example, take the first set to be all the people in the world, the second set as all positive numbers, and use the rule, “Assign to each person his or her exact weight.” This is a function. We have two sets and a rule which assigns to every element in the first set (people) exactly one element in the second set (his or her weight).

## 商科代写|商业数学代写business mathematics代考|MATHEMATICAL FUNCTIONS

We know that a function consists of three components: a domain, a range, and a rule. The domain and range can be any two sets (people, cars, colors, numbers, etc.), while the rule can be given in a variety of ways (arrows, tables, words, etc.). In business situations, the primary concern is with sets of numbers (representing price, demand, advertising expenditures, cost, or profit, etc.) and rules defined by mathematical equations.

At first glance, it may seem strange to think of an equation as a rule, but it is. Consider two identical sets of real numbers and the equation $y=15 x+10$, where $x$ represents a number in the domain and $y$ represents a number in the range. The equation is nothing more than the rule “Multiply each element in

the domain by 15 and add 10 to the result.” Similarly, the equation $y=x^{2}-7$ is the rule “Square each element in the domain and then subtract 7 from the result.”

Whenever we have two sets of numbers and a rule given by an equation, where the variable $x$ denotes an element in the domain and the variable $y$ denotes an element in the range, we simply say that $y$ is a function of $x$ and write $y=f(x)$, although symbols other than $x$ and $y$ labels are frequently used when they are more appropriate to a particular problem.

## 商科代写|商业数学代写business mathematics代考|POLYNOMIAL FUNCTIONS

Mathematical functions are the core of real-world applications and essential tools for decision makers. In this section, we present a class of one of the most useful business functions – polynomials
A function $f(x)$ is a polynomial function if it has the form
$$f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}$$
here $a_{n}, a_{n-1}, \cdots, a_{2}, a_{1}$, and $a_{0}$ are all known numbers.
The powers of $x$ (that is, $n, n-1$, and so on) are required to be nonnegative integers 1 , and the highest power with a corresponding non-zero coefficient a, is called the degree of the polynomial function. The lead coefficient, $a_{n}$, is the coefficient of the $x$ term with the highest power and cannot be zero, but any of the other following coefficients can be. The constant term $a_{0}$ is the coefficient of $x^{0}=1$.

Example 1 Determine which of the following functions are polynomial functions. For those that are, state their degree, and coefficients.
a. $f(x)=3 x^{2}-2 x$
b. $f(x)=0.8 x^{5}-2.25 x^{3}-\sqrt{7}$
c. $f(x)=\sqrt{x}$
d. $f(x)=\frac{1}{3}$
e. $f(x)=\frac{1}{x}$
Solution
a. This is a polynomial function of degree 2 , with $a_{2}=3, a_{1}=-2$, and $a_{0}=0$.
b. This is a polynomial function of degree 5 , with $a_{5}=0.8, a_{4}=0, a_{3}=2.25$. $a_{2}=a_{1}=0$, and $a_{0}=\sqrt{7}$.
c. This is not a polynomial function because $\sqrt{x}=x^{\frac{1}{2}}$. Here $x$ is raised to the $\frac{1}{2}$ power, which is not a non-zero integer.
d. This is a polynomial function of degree 0 , with $a_{0}=\frac{1}{3}$.
e. This is not a polynomial function because $\frac{1}{x}=x^{-1}$. Here $x$ is raised to the $-1$ power, which is a negative integer (this is a rational function, as presented at the end of this section).

## 商科代写|商业数学代写business mathematics代考|CONCEPT OF A FUNCTION

（2）第二组（可能是数字），以及（3）

## 商科代写|商业数学代写business mathematics代考|POLYNOMIAL FUNCTIONS

F(X)=一个nXn+一个n−1Xn−1+⋯+一个2X2+一个1X+一个0

C。F(X)=X
d。F(X)=13

C。这不是多项式函数，因为X=X12. 这里X被提升到12幂，它不是一个非零整数。
d。这是一个 0 次多项式函数，其中一个0=13.
e. 这不是多项式函数，因为1X=X−1. 这里X被提升到−1幂，它是一个负整数（这是一个有理函数，如本节末尾所示）

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## MATLAB代写

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