5.13 摘要:关系和职能

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  摘要:关系和职能

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

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  Identifying a Relation That Is Also a Function  
  ::确定关系是否也是一种职能

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  • Relations can be expressed in various forms, including tables, sets of ordered pairs, pictures, graphs, words, and formulas
    ::各种关系可以以各种形式表达,包括表格、一对定购的配对、图片、图表、单词和公式。
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  • Functions are relations where each output corresponds to a unique input.
    ::功能是指每项产出与独特投入相对应的关系。
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  • For a graph, you can use the Vertical Line Test: Draw vertical lines through the graph. If any line intersects the graph in more than one point, then the graph is not the graph of a function.
    ::对于图形,您可以使用垂直线测试:在图形中绘制垂直线。如果任何线将图形交叉到不止一个点,则图形不是函数的图形。

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  Finding the Domain and Range of a Function 
  ::查找函数的域域和范围

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  • The domain is the set of possible inputs; the range is the set of possible outputs.
    ::域是一套可能的输入;范围是一套可能的输出。
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  • On a graph, you can draw vertical and horizontal lines to help you determine the domain and range, respectively.
    ::在图形中,您可以绘制垂直线和水平线,帮助您分别确定域和范围。
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  • To find the domain of a function that is a fraction, set the denominator equal to 0 to find values to rule out of the domain.
    ::要找到函数分数的域, 设置分母等于 0 的分母, 以找到要排除域的值 。
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  • To find the domain of a function with a square root, set the radicand greater than or equal to 0 to find the values that should be in the domain. 
    ::要找到带平方根的函数域, 请设定值大于或等于 0 的弧度, 以找到域内的值 。

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  Evaluating a Function
  ::评价职能

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  • Replace the variable in the rule with the expression inside the parentheses. 
    ::将规则中的变量替换为括号内的表达式。

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  Determining if a Graph Is Increasing, Decreasing, or Constant
  ::确定图表是否在增加、减少或常数

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  • Read the graph from left to right. It is increasing if the  y- values go up, decreasing if the  y- values go down, and constant if the  y- values stay the same.
    ::从左到右读取图形。如果 Y 值上升,它会增加,如果y 值下降,则会减少,如果y 值保持不变,则会保持不变。

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  Determine the Extrema on the Graph
  ::确定图形上的Extrema

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  • Global or absolute extrema are the highest and lowest points on the entire graph. 
    ::全球或绝对矩形是整个图中最高和最低的点。
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  • Local or relative extrema are the highest or lowest points on an interval of the graph. 
    ::局部或相对的extrema是图间距上的最高或最低点。

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  Transformations of Functions
  ::函数转换

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  • Vertical shift: Add or subtract a value from $\begin{array}{r}f(x)\end{array}$ .
    ::垂直移动: 从 f(x) 中添加或减去值。
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  • Horizontal shift: Add or subtract a value from x .
    ::水平移动:从 x 中添加或减去一个值。
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  • Reflect about x -axis: Negate $\begin{array}{r}f(x)\end{array}$ .
    ::反射 X 轴 : Negate f( x) 。
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  • Reflect about y -axis: Negate x .
    ::反射 Y 轴: Negate x 。
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  • Dilation: Multiply $\begin{array}{r}f(x)\end{array}$  by a constant.
    ::膨胀: 乘以 f(x) 值乘以一个常数 。
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  • The order to perform the transformations in is: reflections, dilations, horizontal shifts, and, lastly, vertical shifts.
    ::进行变换的顺序是:反射、推算、水平移动,最后是垂直转变。

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  Direct, Indirect, Combined, and Joint Variation
  ::直接、间接、合并和联合变化

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  • The direct variation equation is $\begin{array}{r}y=k{x}^n\end{array}$  where $\begin{array}{r}k\ne 0\end{array}$  and $\begin{array}{r}n>0\end{array}$ . 
    ::直接变异方程式是 y=kxn, k0 和 n>0 。
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  • The indirect variation equation is  $\begin{array}{r}y=\frac{k}{x^n}\end{array}$  where  $\begin{array}{r}k\ne 0\end{array}$  and  $\begin{array}{r}n>0\end{array}$ .
    ::间接变异方程式是 y=kxn, k0 和 n>0 。
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  • Combined variation is a mix of direct and indirect variation.
    ::合并变异是直接和间接变异的混合体。
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  • The joint variation equation is $\begin{array}{r}z=k{x}^m{y}^n\end{array}$   where  $\begin{array}{r}k\ne 0\end{array}$  and  $\begin{array}{r}m>0,n>0\end{array}$ . 
    ::联合变异方程式是 z=kxmyn, k0 和 m>0,n>0 。

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

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  In this chapter, we discussed the basic properties of relations, and more specifically functions. As we have seen, functions can be used to model various situations in every day life. In the following chapters, we will discuss several types of functions, including polynomial, rational, radical, exponential, and logarithmic functions. 
  ::在本章中,我们讨论了关系的基本特性,更具体地说,各种功能。正如我们所看到的那样,各种功能可以用来在日常生活中模拟各种情形。在以下各章中,我们将讨论几种功能,包括多面性、理性性、激进性、指数性和对数性功能。

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  Chapter Review
  ::回顾章次审查