### 数学代写|微积分代写Calculus代写|Functions Review

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

## 数学代写|微积分代写Calculus代写|Vertical line test

Not all equations have graphs that represent functions. To be a function, there must be only one output for any given input. Because inputs

are horizontal coordinates and outputs are vertical coordinates, for any horizontal coordinate (input), there can be only one vertical coordinate (output). In other words, no vertical line should intersect the graph at more than one point; this is called the vertical line test.
VERTICAL LINE TEST
Given a graph, if there is a vertical line that intersects the graph at more than one point, then the graph is not the graph of a function. If there is no such line, then the graph is the graph of a function.
Example 4 For each graph in figure 5, is the graph the graph of a function?
Figure 5 The graphs for example 4
Solution (a) Any vertical line (in gray in figure 6) intersects the graph at, at most, one point. The graph passes the vertical line test, so the graph does represent the graph of a function.
(b) There is a vertical line that intersects the graph at more than one point (see figure 7 ), so the graph fails the vertical line test. The graph does not represent the graph of a function.

## 数学代写|微积分代写Calculus代写|Function values

To evaluate $f(x)$ at $x=4$, we simply replace $x$ in the output expression with 4 , wherever the $x$ appears. The same is true for any input; we replace every appearance of $x$ with the given input.

Example 5 For $f(x)=x^{2}+2$, find $(a) f(4),(b) f(4+h),(c) f(\sqrt{x})$, and
(d) $\frac{7+f(x)}{f(3)}$.
Solution (a) To find $f(4)$, we replace $x$ with 4 (emphasized using bold):
\begin{aligned} &f(x)=x^{2}+2 \ &f(4)=4^{2}+2=18 \end{aligned}
(b) To find $f(4+h)$, we replace $x$ with $4+h$. Because of the manner in which we write expressions, if the input expression has more than one term, it should be surrounded by parentheses:

\begin{aligned} f(x) &=x^{2}+2 \ f(4+h) &=(4+h)^{2}+2 \ &=16+8 h+h^{2}+2 \ &=18+8 h+h^{2} . \end{aligned}
(c) Again, the replacement is emphasized using bold:
\begin{aligned} f(x) &=x^{2}+2 \ f(\sqrt{x}) &=(\sqrt{x})^{2}+2 \ &=x+2 \end{aligned}
(d) Function values can sit anywhere in an expression. They are evaluated and placed in the expression where written:
\begin{aligned} \frac{7+f(x)}{f(3)} &=\frac{7+\left(x^{2}+2\right)}{3^{2}+2} \ &=\frac{x^{2}+9}{11} . \end{aligned}
If $f(x)$ contains more than one occurrence of the variable $x$, we replace each $x$ with the input expression.
Example 6 For $f(x)=\frac{x^{2}-5 x+2}{x-7}$, find $f(3)$.
Solution We replace each occurrence of $x$ with 3 , emphasized in bold:
\begin{aligned} f(3) &=\frac{3^{2}-5 \cdot 3+2}{3-7} \ &=\frac{-4}{-4} \ &=1 . \end{aligned}

## 数学代写|微积分代写Calculus代写|Finding domains and ranges

Unless otherwise specified, when we are asked for the domain of a function, this means the natural domain, the set of all possible acceptable inputs. What must be avoided are operations that are not

defined (at least for real numbers), such as division by zero or square roots of negative numbers. As our list of functions grows, our list of operations that must be avoided may also grow (for instance, logarithms are defined only for positive numbers); but, for now, our list of what to look for is short.

Finding the range of a function is often not nearly as straightforward algebraically as finding the domain, and, as a result, we often resort to checking the graph of the function to determine the function’s range.
Example 7 Find the domain and range of $f(x)=x^{2}+7$
Solution The expression $x^{2}+7$ is defined for every real number. Any number can be squared; we can add 7 to any number. There is no division by zero, no square root of a negative number, nothing that must be avoided. The domain of $f$ is the set of all real numbers, which may be expressed as $\mathbf{R}$ or as $(-\infty, \infty)$.

Checking the graph (figure 8 ), we see that the $y$-coordinates are 7 or larger. Therefore the range is $[7, \infty)$, which may also be written in inequality form as $y \geq 7$.

## 数学代写|微积分代写Calculus代写|Vertical line test

(b) 有一条垂直线在多个点与图形相交（见图 7），因此图形未通过垂直线测试。图不代表函数的图。

## 数学代写|微积分代写Calculus代写|Function values

(d)7+F(X)F(3).

F(X)=X2+2 F(4)=42+2=18
(b) 寻找F(4+H), 我们替换X和4+H. 由于我们编写表达式的方式，如果输入表达式有多个项，则应该用括号括起来：

F(X)=X2+2 F(4+H)=(4+H)2+2 =16+8H+H2+2 =18+8H+H2.
(c) 同样，替换用粗体强调：

F(X)=X2+2 F(X)=(X)2+2 =X+2
(d) 函数值可以位于表达式中的任何位置。它们被评估并放置在写入的表达式中：

7+F(X)F(3)=7+(X2+2)32+2 =X2+911.

F(3)=32−5⋅3+23−7 =−4−4 =1.

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。