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OSSD课程之什么是函数以及函数的特征

2020-05-18 环球教育

  大家好,今天我们来聊一下什么是函数以及函数的特征。

           A function is a special type of relation. A function is a set of ordered pairs in which for every value of x, there is only one value of y, that is no two ordered pairs have the same first element.

  函数是一种特殊的关系类型。函数是一组有序对,其中对于 x 的每个值,只有一个值为 y,即没有两个有序对具有相同的第一个元素。

  举一个例子大家就能够清晰地明白什么是函数,什么不是函数。例如:

  The relation {(9, 81), (10, 100), (11, 121), (12, 144)} is a function because for each value of x, there is only one value of y.

  The relation {(9, 81), (10, 100), (11, 121), (12, 144), (12, 156)} is not a function because when x = 12, y can be equal to 144 or 156.

  关系{(9,81),(10,100),(11,121),(12,144)}是一个函数,因为对于x的每个值,只有y的一个值。

  关系{(9,81),(10,100),(11,121),(12,144),(12,156)}不是函数,因为当x = 12时,y可以等于144或156。

  函数中又有许多名词,我们逐一来看一下:

  1. Words:

  E.g.  Each output value is equal to the square of each input value.

  每个输出值等于每个输入值的平方

  2. Equations:

  E.g.  g( x ) = x ²    (The name of this function is “g”, the input is written in the brackets and the output is written beside the equal sign)

  g( x ) = x ²  (此函数的名称为“ g ”,输入写在方括号中,输出写在等号旁边)

  3. Tables of Values:

  

The input is the first row or column of a table (the independent variable) and the output is the second row or column of a table (the dependent variable since it depends on the input).

  输入是表格的第一列(自变量),输出是表格的第二列(因变量,因为它取决于输入)

  4. Mapping Diagrams:

  

The set of input and output numbers are in their own separate groupings and each input value is connected to its one output value by an arrow (which only points from input to output, uni-directional).

  输入和输出编号的集合在它们自己的单独分组中,并且每个输入值通过箭头(仅从输入指向输出,单向指向)连接到其一个输出值。

  5. Data Sets:

  

The first value of each ordered pair is the input which produces only one output (the second value from each ordered pair) and all of the ordered pairs are separated by commas and grouped inside of curly brackets .

  每个有序对的第一个值是输入,该输入仅产生一个输出(每个有序对的第二个值),所有有序对用逗号分隔并放在大括号内

  6. Graphs:

  


      The input variables are found along the horizontal axis and the output variables are found along the vertical axis.

  沿水平轴找到输入变量,沿垂直轴找到输出变量。

  我们可以通过以下方法来判断一种关系是不是函数:

       One way to determine if a relation is a function is to graph the relation and then use the vertical line test. If any vertical line passes through more than one point on the graph, then the relation is not a function.

  确定关系是否为函数的一种方法是绘制关系图,然后使用垂直线测试。如果任何垂直线穿过图形上的多个点,则该关系不是函数。

  

  

  In left graph, the relation is a function. No vertical line passes through more than one point.

  In right graph, the relation is not a function. The vertical line shown passes through two points, so there are two values of y for the same value of x.

  我们再举一个例子:

  Most equations isolated as “y=” are functions:          

  

  Equations that have ‘‘y2 ’’are NOT functions:

  

   通过这个例子,我们便可以判断出什么是函数什么不是函数,由于上面的式子每一个输入仅对应一个输出,所以是函数,而下面的函数每一个输入对应两个输出,那么这个式子便不是函数。


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