9.12 摘要:合理职能

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• Select section 摘要: 合理函数

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  摘要: 合理函数

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  In this chapter, we learned about: 
  ::在本章中,我们了解到:

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  Simplifying Rational Expressions
  ::简化逻辑表达式

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  • F actor the polynomials in the numerator and the denominator, and then cancel any common factors. 
    ::乘以分子和分母中的多数值,然后取消任何共同系数。

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  Operations With Rational Expressions 
  ::有理性表达式的操作

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  • Multiplication: We multiply the numerators together and the denominators together. Factor the rational expressions before multiplying to make simplifying easier.
    ::乘法 : 我们将分子和分母一起乘。 乘法前先将理性表达法乘以方便简化 。
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  • Division: Multiply by the reciprocal.
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    • Complex fractions: If the complex fraction has one fraction in the numerator and one fraction in the denominator, rewrite as a division problem.
      ::复杂分数:如果复杂分数在分子中有一分数,分母中有一分数,则重写为分裂问题。
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    • If there is addition or subtraction between the fractions in the numerator or the denominator, you can find a common denominator for all the fractions and multiply the numerator and denominator by that common denominator, or you can find a common denominator for the top and a common denominator for the bottom and then rewrite as a division problem.
      ::如果分子或分母中的分数有增减,您可以为所有分数找到一个共同的分母,并将分子和分母乘以该共同分母,或者为顶部找到一个共同的分母,为底部找到一个共同的分母,然后作为一个分裂问题重写。
    
    ::乘以对等分数 。 复杂分数 : 如果复杂分数在分子中有一个分数, 在分母中有一个分数, 则重写为分数问题 。 如果在分子或分母中分数之间的增加或减法, 您可以为所有分数找到一个共同分母, 并将分子和分母乘以该共同分母, 或者您可以为顶部找到一个共同分母, 底部找到一个共同分母, 然后作为一个分数问题重写 。
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  • Addition and Subtraction:
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    • Like Denominators: Add or subtract the numerators and combine over a common denominator. Simplify the result if possible.
      ::类似分母 : 添加或减去计数器, 并结合到一个共同分母上。 如果可能, 简化结果 。
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    • U nlike Denominators: Find the LCD, create equivalent fractions, replace the original fraction with its equivalent form, combine, and simplify.
      ::与排减器不同:查找液晶显示器,创造等效分数,以等效形式替换原分数,合并和简化。
    
    ::增量和减量: 类似分母: 添加或减去计数器, 并组合在一个共同分母之上。 如果可能的话, 简化结果 。 不同于分母 : 查找液晶体, 创建等效分数, 以等效形式替换原始分数, 组合和简化 。

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  How to Solve Rational Equations
  ::如何解析合理等式

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  • Using the LCD: F ind the LCD, multiply both sides of the equation by the LCD, solve the equations, and check for extraneous solutions.
    ::使用 LCD: 查找 LCD, 将方程的两边乘以 LCD, 解析方程, 并检查不相干的解决办法 。
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  • By cross-multiplication: Multiply the numerator of one fraction with the denominator of the other fraction and set the expressions equal to each other. Solve the equation and check for extraneous solutions.
    ::通过交叉乘法:将一个分数的分子乘以另一个分数的分母,并将表达式设为等号。解析方程并检查不相干的解决办法。

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  How to Graph Rational Functions
  ::如何图形逻辑函数

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  • To find the domain of a rational function, set the denominator equal to zero and solve the equation. The solutions should be eliminated from the domain of the function.
    ::要找到一个合理函数的域, 将分母设置为零, 并解析等式 。 解决方案应该从函数的域中删除 。
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  • The vertical asymptote of a reciprocal function in standard form is $\begin{array}{r}x=h,\end{array}$  and the horizontal asymptote is $\begin{array}{r}y=k\end{array}$ .
    ::标准格式对等函数的垂直时点为 x=h,水平时点为 y=k。
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  • To graph a reciprocal function in standard form, determine the domain of the function (this will also be the location of the vertical asymptote), find the horizontal asymptote, and create a table of values with some values to the right of the vertical asymptote and some to the left of the vertical asymptote.
    ::要用标准格式绘制对等函数,请确定函数的域(这还将是垂直静态的位置),找到水平静态,并创建含有垂直静态右侧和垂直静态左侧某些值的数值表。
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  • To find a vertical asymptote or removable discontinuities of a rational function, factor the numerator and the denominator. There is a vertical asymptote when you CANNOT cancel the factor in the denominator with a factor of the numerator. There is a removable discontinuity when you CAN cancel the factor in the denominator with a factor of the numerator.
    ::要找到一个理性函数的垂直零点或可移动的不连续性,请乘以分子和分母。当您用分子因子取消分母中的系数时,CANNOT会有一个垂直的零点。如果您可以用分子因子取消分母中的系数,则会有一个可移动的不连续性。
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  • If $\begin{array}{r}m<n\end{array}$ , then there is a horizontal asymptote at $\begin{array}{r}y=0\end{array}$ .
    ::如果 m < n, 那么y=0 就会有一个水平同位数 。
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  • If $\begin{array}{r}m=n\end{array}$ , then there is a horizontal asymptote at $\begin{array}{r}y=\frac{a_m}{b_n}\end{array}$ (ratio of the leading coefficients).
    ::如果 m=n, y= ambn (主要系数的比值) 就会有一个水平的同位数 。
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  • If $\begin{array}{r}m=n+1\end{array}$ , then there is a oblique (slant) asymptote at $\begin{array}{r}y=(a_m{x}^m+\dots +a_0)÷(b_n{x}^n+\dots +b_0)\end{array}$ without the remainder. A good estimate is  $\begin{array}{r}y=\frac{a_m}{b_n}x\end{array}$ .
    ::如果 m=n+1, 那么y= (amxm+...+a0)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Y=\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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  • If  $\begin{array}{r}m>n+1\end{array}$ , then there are  oblique asymptotes.
    ::如果 m>n+1, 则有倾斜的单位数 。
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  • To graph a rational function, find the domain of the function. Find any vertical asymptotes or removable discontinuities. Find any horizontal asymptotes or oblique asymptotes. Find any intercepts. Create a table of values to fill out the graph.
    ::要绘制一个合理函数,请找到函数的域。找到任何垂直的单位或可移动的不连续状态。找到任何水平的单位或倾斜的单位。查找任何拦截。创建一个数值表以填充图形。

   

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  Looking Back, Looking Forward
  ::回顾,展望未来

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  In this chapter, we covered rational expressions and rational functions. These can be used when finding the resistance of a circuit, determining the load on a bridge, or showing how division of work can get a job done faster.  
  ::在本章中,我们涵盖了理性表达和理性功能,这些功能可用于寻找电路阻力、确定桥梁负荷或显示分工如何能更快地完成工作。

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  We have covered the functions that involve the four common operations —addition, subtraction, multiplication, and division. Next we look at functions that involve radicals, such as  square roots.   
  ::我们承担了涉及四个共同行动的职能——增加、减法、乘法和划分。 接下来我们审视的是涉及激进分子的职能,比如平底。

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  Chapter Review
  ::回顾章次审查