### 计算机代写|计算机图形学作业代写computer graphics代考|Function Domains and Ranges

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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

## 计算机代写|计算机图形学作业代写computer graphics代考|Function Domains and Ranges

The following descriptions of domains and ranges only apply to functions with one independent variable: $f(x)$.
Returning to the above function:
$$y=f(x)=3 x^{2}+2 x+4$$
the independent variable $x$, can take on any value from $-\infty$ to $\infty$, which is called the domain of the function. In this case, the domain of $f(x)$ is the set of real numbers

$\mathbb{R}$. The notation used for intervals, is also used for domains, which in this case is
$$]-\infty, \infty[$$
and is open, as there are no precise values for $-\infty$ and $\infty$.
As the independent variable takes on different values from its domain, so the dependent variable, $y$ or $f(x)$, takes on different values from its range. Therefore, if the domain of the linear function $f(x)=3 x+4$ is $[-4,4]$, the range of $f(x)$ is calculated by finding $f(-4)$ and $f(4)$ :
$$\begin{gathered} f(-4)=-12+4=-8 \ f(4)=12+4=16 \end{gathered}$$
and the range is $[-4,4]$.
Although calculating the range of linear functions is simple, other types of functions require a knowledge of calculus.
The domain of $\log x$ is
] $0, \infty[$
which is open, because $x \neq 0$. Whereas, the range of $\log x$ is
$$]-\infty, \infty[.$$
The domain of $\sqrt{x}$ is
$[0, \infty[$
which is half-open, because $\sqrt{0}=0$, and oo has no precise value. Similarly, the range of $\sqrt{x}$ is
$[0, \infty[.$
Sometimes, a function is sensitive to one specific number. For example, in the function
$$y=f(x)=\frac{1}{x-1}$$
when $x=1$, there is a divide by zero, which is meaningless. Consequently, the domain of $f(x)$ is the set of real numbers $\mathbb{R}$, apart from $1 .$

## 计算机代写|计算机图形学作业代写computer graphics代考|Odd and Even Functions

An odd function satisfies the condition:
$$f(-x)=-f(x)$$

where $x$ is located in a valid domain. Consequently, the graph of an odd function is symmetrical relative to the origin. For example, $\sin (\theta)$ is odd because
$$\sin (-\theta)=-\sin \theta$$
as illustrated in Fig. 3.6. Other odd functions include:
\begin{aligned} &f(x)=a x \ &f(x)=a x^{3} \end{aligned}
An even function satisfies the condition:
$$f(-x)=f(x)$$
where $x$ is located in a valid domain. Consequently, the graph of an even function is symmetrical relative to the $f(x)$ axis. For example, $\cos \theta$ is even because
$$\cos (-\theta)=\cos \theta$$
as illustrated in Fig. 3.7. Other even functions include:
\begin{aligned} &f(x)=a x^{2} \ &f(x)=a x^{4} \end{aligned}

## 计算机代写|计算机图形学作业代写computer graphics代考|Power Functions

Functions of the form $f(x)=x^{n}$ are called power functions of degree $n$ and are either odd or even. If $n$ is an odd natural number, then the power function is odd, else if $n$ is an even natural number, then the power function is even.

The above description of algebra should be sufficient for the reader to understand the following chapters. However, one should remember that this is only the beginning of a very complex subject.

## 计算机代写|计算机图形学作业代写computer graphics代考|Function Domains and Ranges

R. 用于区间的符号也用于域，在这种情况下是
]−∞,∞[

F(−4)=−12+4=−8 F(4)=12+4=16

]0,∞[

]−∞,∞[.

[0,∞[

[0,∞[.

## 计算机代写|计算机图形学作业代写computer graphics代考|Odd and Even Functions

F(−X)=−F(X)

F(X)=一种X F(X)=一种X3

F(−X)=F(X)

F(X)=一种X2 F(X)=一种X4

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