Course: 2.4 对称 | The Way To Learn

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Introduction
::导言

Function is useful when analyzing functions because it provides powerful information about a mathematical model's behavior. Consider this example of an apartment building with 50 rental units, each unit initially costing $580 a month to rent. The apartment manager knows that every time the rent is increased by $20, one tenant moves out. The function $\begin{aligned} R (x) = (580 + 20 x) (50 - x) \end{aligned}$ can be used to calculate the amount of rent that is collected based on the number of $20 rent increases. The domain of this function is $\begin{aligned} 0 \leq x \leq 50 \end{aligned}$ , because there are 50 apartments in the building. The graph of $\begin{aligned} R (x) \end{aligned}$ has an absolute maximum value of about $31,000 rent near 10 rent increases. After that point, the total amount of rent collected decreases as each tenant's rent increases. The manager can use the symmetry about the line near $\begin{aligned} x = 10 \end{aligned}$ to inform her decisions when making increases near there.
::函数在分析函数时有用, 因为它提供了关于数学模型行为的有力信息。请参考一个有50套出租单元的公寓大楼的例子, 每个单元最初每月租金为580美元。公寓经理知道, 每次租金增加20美元, 都会有一个租户搬走。 R( x) =( 580+20x) ( 50 - x) 可以用来计算根据20美元租金增加数收取的租金数额。这个函数的域是 0x=50, 因为有50套公寓。 R( x) 图表的绝对最高值约为31 000美元租金, 接近10套租金增加数。在此点之后, 收取的租金总额随着每个租户的租金增加而减少。经理可以使用关于接近 x=10的行数的对称来计算租金数额。当她在该楼附近增加租金时, 此函数的域是 0x=50 。

Even and odd function analysis formalizes the algebraic definitions of symmetry. Even functions exhibit symmetry with respect to the $\begin{aligned} y \end{aligned}$ -axis or reflection symmetry. Odd functions have symmetry with respect to the origin or rotational symmetry about the origin. "With respect to the origin" means the graph of the function is reflected across both the $\begin{aligned} x - \end{aligned}$ and $\begin{aligned} y \end{aligned}$ -axes.
::偶数和奇数函数分析使对称的代数定义正式化。偶数函数在 y 轴或反射对称方面表现出对称性。奇数函数在起源或对源的旋转对称性方面具有对称性。 “ 关于起源” 表示函数的图表在 x - 和 y 轴上都反映。

Even and Odd Function s
::偶函数和奇函数

Functions that display symmetry with respect to $\begin{aligned} x = 0 \end{aligned}$ (the $\begin{aligned} y \end{aligned}$ -axis) are called even functions .
::相对于 x=0 (y-轴) 显示对称函数的函数被称为偶函数。

$\begin{aligned} f (x) \end{aligned}$ is even only if $\begin{aligned} f (- x) = f (x) \end{aligned}$ for every $\begin{aligned} x \end{aligned}$ in the domain
::f( x) 仅在域内每 x x 的 f( x) =f( x) 时才为偶数

Functions that have rotational symmetry about the origin are called odd functions .
::函数对源的旋转对称被称为奇函数。

$\begin{aligned} f (x) \end{aligned}$ is odd only if $\begin{aligned} f (- x) = - f (x) \end{aligned}$ for every $\begin{aligned} x \end{aligned}$ in the domain.
::f( x) 仅在域内每 x 个 f 的 f -x)\\\\\\ f( x) 时才是奇数。



Play, Learn, and Explore rotational symmetry with odd functions:
::播放、学习和探索与奇函数的旋转对称 :

Examples
::实例

Example 1
::例1

Show that $\begin{aligned} f (x) = 3 x^{4} - 5 x^{2} + 1 \end{aligned}$ is even.
::显示 f( x) = 3x4 - 5x2+1 是相等的。

Solution:
::解决方案 :

Step 1: Replace $\begin{aligned} x \end{aligned}$   with $\begin{aligned} - x \end{aligned}$ :

$\begin{aligned} f (- x) & = 3 (- x)^{4} - 5 (- x)^{2} + 1 \\ = 3 x^{4} - 5 x^{2} + 1 \\ = f (x) \end{aligned}$

::第1步:将x改为-x: f(-x)=3(-x)4-5(-x)2+1=3x4-5x2+1=5x2+1=f(x)

Step 2: Compare with the original function. Since $\begin{aligned} f (- x) = f (x) \end{aligned}$ , $\begin{aligned} f (x) \end{aligned}$ is an even function.
::第2步:与原始函数相比。因为 f( - x) =f( x), f( x) 是一个偶数函数。

Example 2
::例2

Show that $\begin{aligned} f (x) = 4 x^{3} - x \end{aligned}$ is odd.
::显示 f( x) =4x3- x 异常。

Solution:
::解决方案 :

Step 1: Replace $\begin{aligned} x \end{aligned}$ with $\begin{aligned} - x : \end{aligned}$

::第1步:将x替换为-x:

$\begin{aligned} f (- x) & = 4 (- x)^{3} - (- x) \\ = - 4 x^{3} + x \\ = - (4 x^{3} - x) \\ = - f (x) \end{aligned}$

:- x) = 4(- x) 3 - (- x) 4x3+x (-4x3- x) f( x)

Step 2: Compare with the original function. Since $\begin{aligned} f (- x) = - f (x) \end{aligned}$ , $\begin{aligned} f (x) \end{aligned}$ is an odd function.
::第2步:与原始函数相比。因为 f-x) f( x), f( x) 是一个奇数函数。

Example 3
::例3

Identify whether the function is even, odd or neither and explain why:
::确定该函数是偶数、奇数还是两者兼有,并解释原因:

$\begin{aligned} f (x) = 4 x^{3} - | x | \end{aligned}$
:xx) = 4x3

Solution
::解决方案

Step 1: Replace $\begin{aligned} x \end{aligned}$ with $\begin{aligned} - x \end{aligned}$ :
::第1步:将x改为-x:

$\begin{aligned} f (- x) & = 4 (- x)^{3} - | - x | \\ = - 4 x^{3} - | x | \end{aligned}$

:-x)=4(-x)3-x)4x4x3xxxxxx

Step 2: Compare with the original function. Since $\begin{aligned} f (- x) \neq f (x) \end{aligned}$ the function is not even. Since    $\begin{aligned} f (- x) \neq - f (x) = - 4 x^{3} + | x | \end{aligned}$ , the function is not odd. Therefore , this function is neither even nor odd.
::第2步 : 与原始函数比较。由于 f =x) =\\ f( x) 函数不相等。由于 f =x) =\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Example 4
::例4

Which of the following basic functions are even, which are odd, and which are neither?
::以下哪些基本功能是偶数,哪些是奇数,哪些两者都不是?

$\begin{aligned} \begin{aligned} f (x) & = x, f (x) = x^{2}, f (x) = x^{3}, f (x) = | x |, f (x) = \frac{1}{x}, f (x) = \sqrt{x} \\ f (x) & = \sin x, f (x) = \cos x, f (x) = a b^{x}, and f (x) = l o g_{b} x \end{aligned} \end{aligned}$
::f(x)=xxxxxxxxxxxxxxxxxxx2, f(x)x=x3, f(x)xxxxx}f(x)=1xx、 f(x)xxxxxxx、 f(x)=cos*xx、 f(x)=abx和 f(x)=logbx

Solution
::解决方案

Even Functions: The quadratic function $\begin{aligned} f (x) = x^{2} \end{aligned}$ , the absolute value function $\begin{aligned} f (x) = | x | \end{aligned}$ , and the cosine function $\begin{aligned} f (x) = \cos x \end{aligned}$ are even.
::偶数函数 : 二次函数 f( x) =x2, 绝对值函数 f( x) x , 余弦函数 f( x) =cos x 均相等。

Odd Functions: The identity or linear function $\begin{aligned} f (x) = x \end{aligned}$ , the cubic function $\begin{aligned} f (x) = x^{3} \end{aligned}$ , the reciprocal function $\begin{aligned} f (x) = \frac{1}{x} \end{aligned}$ , and the sine function $\begin{aligned} f (x) = \sin x \end{aligned}$ are odd.
::函数 : 身份或线性函数 f( x) =x, 立方函数 f( x) =x3, 互惠函数 f( x) = 1x, 和正弦函数 f( x) = sinx 是奇数。

Neither: The square root function $\begin{aligned} f (x) = \sqrt{x} \end{aligned}$ , the exponential function $\begin{aligned} f (x) = a b^{x} \end{aligned}$ , and the log function $\begin{aligned} f (x) = \log_{b} x \end{aligned}$ are neither. As a note, the logistic function is also neither because it is rotationally symmetric about the point $\begin{aligned} (0, \frac{1}{2}) \end{aligned}$ as opposed to the origin.
::都不是 : 平方根函数 f( x) =x, 指数函数 f( x) =abx, 和日志函数 f( x) = logb* x 都不是。注意, 后勤函数也不是, 因为它对点( 0 , 12) 与源值是旋转对称的。

Example 5
::例5

The mechanics of combining functions is covered in another section, but let's begin by thinking about how operations might affect the symmetry of the resulting function.
::合并功能的机理在另一部分中涉及, 但是让我们先考虑一下操作如何影响所产生的功能的对称性。

Suppose $\begin{aligned} h (x) \neq 0 \end{aligned}$ is an even function and $\begin{aligned} g (x) \neq 0 \end{aligned}$ is an odd function. Let $\begin{aligned} f (x) = h (x) + g (x) . \end{aligned}$ Is $\begin{aligned} f (x) \end{aligned}$ even or odd?
::假设 h( x) @% 0 是一个偶数函数, g( x)\% 0 是一个奇数函数。 Let f( x)=h( x)+g( x). f( x) 是偶数还是奇数?

Solution:
::解决方案 :

If $\begin{aligned} h (x) \end{aligned}$ is even, then $\begin{aligned} h (- x) = h (x) \end{aligned}$ . If $\begin{aligned} g (x) \end{aligned}$ is odd, then $\begin{aligned} g (- x) = - g (x) \end{aligned}$ . Therefore, $\begin{aligned} f (- x) = h (- x) + g (- x) = h (x) - g (x) \end{aligned}$ .
::如果 h( x) 是偶数, 那么h( x) =h( x) 。如果 g( x) 是奇数, 那么g( x) =g( x) 。因此, f( - x) =h( - x) +g( - x) =h( x) =h( x) - g( x) 。

This does not match $\begin{aligned} f (x) = h (x) + g (x) \end{aligned}$ , nor does it match $\begin{aligned} - f (x) = - h (x) - g (x) \end{aligned}$ . So the sum of an even function and an odd function is neither even nor odd.
::此函数不匹配 f( x) =h( x) +g( x) , 也不匹配 - f( x) \\\ ( x) - g( x) 。因此, 偶函数和奇函数的总和既不是偶数,也不是奇数。

Example 6
::例6

Determine whether the following function is even, odd, or neither:
::确定下列函数是偶数、奇数还是偶数:

$\begin{aligned} f (x) = x (x^{2} - 1) (x^{4} + 1) \end{aligned}$
:xx) =x(x2- 1)(x4+1)

Solution:
::解决方案 :

Step 1: Replace $\begin{aligned} x \end{aligned}$ with $\begin{aligned} - x \end{aligned}$ :
::第1步:将x替换为-x:

$\begin{aligned} f (x) & = x (x^{2} - 1) (x^{4} + 1) \\ f (- x) & = (- x) ((- x)^{2} - 1) ((- x)^{4} + 1) \\ = - x (x^{2} - 1) (x^{4} + 1) \\ = - f (x) \end{aligned}$

:x) =x(x2- 1)(x4+1) f(x) =(x)(x) (-x)(x) 2-1 ((-x) 4+1) x(x2- 1)(x4+1)(x4+1) ** f(x)

Step 2: Compare with the original function. The function is odd.
::第2步:与原始函数相比。该函数是奇数。

Summary
::摘要

A function is even if $\begin{aligned} f (- x) = f (x) \end{aligned}$ for every $\begin{aligned} x \end{aligned}$ in the domain. Even functions have reflection symmetry about the vertical line $\begin{aligned} x = 0 \end{aligned}$ (the $\begin{aligned} y \end{aligned}$ -axis).
::A 函数对域中每个 x 的 f( - x) =f( x) 函数是偶数。即使是函数对垂直线 x=0 (y- 轴) 的反射对称。

A function is odd if $\begin{aligned} f (- x) = - f (x) \end{aligned}$ for every $\begin{aligned} x \end{aligned}$ in the domain. Odd functions have rotation symmetry about the origin, which means if you rotate the graph 180 degrees around the point (0, 0), you end up with the same graph.
::函数是奇数,如果 F(- x)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ x\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Even and odd functions describe different types of symmetry, but both derive their name from the properties of exponents. A negative number raised to an even number will always be positive. A negative number raised to an odd number will always be negative.
::偶数函数和奇数函数描述不同类型的对称,但两者都取自推算符的属性。以偶数表示的负数总是正数。以奇数表示的负数总是负数。

Review
::回顾

Determine whether the following functions are even, odd, or neither:
::确定下列功能是偶数、奇数还是两者兼有:

1. $\begin{aligned} f (x) = - 4 x^{2} + 1 \end{aligned}$
::1. f(x)4x2+1

2. $\begin{aligned} g (x) = 5 x^{3} - 3 x \end{aligned}$
::2. g(x)=5x3-3-3x

3. $\begin{aligned} h (x) = 2 x^{2} - x \end{aligned}$
::3. h(x)=2x2-x

4. $\begin{aligned} j (x) = (x - 4) (x - 3)^{3} \end{aligned}$
::4. j(x)=(x-4)(x-3)3

5. $\begin{aligned} k (x) = x (x^{2} - 1)^{2} \end{aligned}$
::5. k(x)=x(x2-1)2

6. $\begin{aligned} f (x) = 2 x^{3} - 5 x^{2} - 2 x + 1 \end{aligned}$
::6. f(x)=2x3-5x2-2-2x+1

7. $\begin{aligned} g (x) = 2 x^{2} - 4 x + 2 \end{aligned}$
::7. g(x)=2x2-4x+2

8. $\begin{aligned} h (x) = - 5 x^{4} + x^{2} + 2 \end{aligned}$
::8. h(x) 5x4+x2+2

9. Suppose $\begin{aligned} h (x) \end{aligned}$ is even and $\begin{aligned} g (x) \end{aligned}$ is odd. Show that $\begin{aligned} f (x) = h (x) - g (x) \end{aligned}$ is neither even nor odd.
::9. 假设h(x)是偶数,g(x)是奇数。显示 f(x)=h(x)-g(x)既不是偶数,也不是奇数。

10. Suppose $\begin{aligned} h (x) \end{aligned}$ is even and $\begin{aligned} g (x) \end{aligned}$ is odd. Show that $\begin{aligned} f (x) = \frac{h (x)}{g (x)} \end{aligned}$ is odd.
::10. 假设h(x)是偶数,g(x)是奇数。显示 f(x)=h(x)g(x)是奇数。

11. Suppose $\begin{aligned} h (x) \end{aligned}$ is even and $\begin{aligned} g (x) \end{aligned}$ is odd. Show that $\begin{aligned} f (x) = h (x) \cdot g (x) \end{aligned}$ is odd.
::11. 假设h( x) 是偶数, g( x) 是奇数。显示 f( x) =h( x) =g( x) 是奇数。

12. Is the sum of two even functions always an even function? Explain.
::12. 两个偶数函数的总和是否总是一个偶数函数?

13. Is the sum of two odd functions always an odd function? Explain.
::13. 两个奇数函数的总和是否总是奇数函数?

14. Why are some functions neither even nor odd?
::14. 为什么有些功能甚至既不奇怪,也不奇怪?

15. If you know that a function is even or odd, what does that tell you about the symmetry of the function?
::15. 如果你知道一个函数是偶数或奇数,那你对函数的对称性有什么看法?

Review (Answers)
::回顾(答复)

Please see the Appendix.
::请参看附录。

Section outline

General

对对称

Introduction ::导言

Even and Odd Function s ::偶函数和奇函数

Examples ::实例

Summary ::摘要

Review ::回顾

Review (Answers) ::回顾(答复)

Introduction
::导言

Even and Odd Function s
::偶函数和奇函数

Examples
::实例

Summary
::摘要

Review
::回顾

Review (Answers)
::回顾(答复)