函数语言
什么是映射?
- 一个映射转换一套价值(输入)转换为另一组值(输出)
- 映射可以是:
- 一对一的
- 每个输入被映射到就是唯一的一个输出
- 没有两个输入映射到相同的输出
- 例如:对输入进行立方体化的映射
- 多对一
- 每个输入被映射到一个输出
- 多个输入可以映射到相同的输出
- 例如:对输入进行平方的映射
- 一对多
- 输入可以映射到不止一个输出
- 没有两个输入映射到相同的输出
- 例如:一个映射,它给出的数字平方等于输入
- 多对多
- 输入可以映射到不止一个输出
- 多个输入可以映射到相同的输出
- 例如:给出输入因子的映射
- 一对一的
什么是函数?
- 一个函数两组数字之间的映射在哪里每个输入映射到只有一个输出
- 输出不需要唯一
- 一对一的而且多对一映射是函数
- 映射是一个函数,如果它的图通过垂直线测试
- 任何垂直的线会与图相交吗最多一次
函数用什么符号?
- 函数用字母表示(例如
" class="Wirisformula" role="math" alt="空间f,空间v,空间g," style="vertical-align:-6px;height:22px;width:57px" loading="lazy">等)f , v , g , {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 函数后面有一个括号中的变量
- 这显示了函数的输入
- 这封信
" class="Wirisformula" role="math" alt="空间f" style="vertical-align:-6px;height:22px;width:14px" loading="lazy">最常用于函数,并将用于本修订说明的其余部分f {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
" class="Wirisformula" role="math" alt="空格f左括号x右括号" style="vertical-align:-6px;height:22px;width:36px" loading="lazy">表示函数值的表达式f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空间f" style="vertical-align:-6px;height:22px;width:14px" loading="lazy">当对变量求值时f {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 函数表示法摆脱了文字的需要,这使得它通用
" class="Wirisformula" role="math" alt="空间f = 5" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">当f = 5 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空间x = 2" style="vertical-align:-6px;height:22px;width:41px" loading="lazy">可以简单地写成x = 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f左括号2右括号等于5" style="vertical-align:-6px;height:22px;width:61px" loading="lazy">f ( 2 ) = 5 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
函数的定义域和值域是什么?
- 的域函数的值是用来表示的值的集合输入
- 域应该用函数来表示
- 如果没有说明一个定义域,则假定该定义域是作为函数输入的所有实值
- 域是根据输入来表示的
" class="Wirisformula" role="math" alt="空格x小于等于2" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">x ≤ 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 的范围函数的值是给定值的集合输出
- 范围取决于域
- 范围是根据输出来表示的
" class="Wirisformula" role="math" alt="空格f有弹性的左括号x右括号大于或等于0" style="vertical-align:-6px;height:22px;width:61px" loading="lazy">f ( x ) ≥ 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 要画出一个函数,我们使用输入为x坐标和输出为y坐标
" class="Wirisformula" role="math" alt="空格f左括号2右括号等于5" style="vertical-align:-6px;height:22px;width:61px" loading="lazy">对应于坐标(2,5)f ( 2 ) = 5 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 把函数画成图形可以帮助你直观地看到范围
- 普通的数字集合有特殊的符号:
" class="Wirisformula" role="math" alt="直接实数" style="vertical-align:-3px;height:19px;width:14px" loading="lazy">表示可以放在数轴上的所有实数ℝ {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="X元素的实数" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">意味着x ∈ ℝ {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">是实数x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
" class="Wirisformula" role="math" alt="直有理数" style="vertical-align:-3px;height:19px;width:16px" loading="lazy">表示所有有理数ℚ {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="A / b" style="vertical-align:-17px;height:47px;width:19px" loading="lazy">在哪里一个而且b都是整数和b≠0a b {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="直整数" style="vertical-align:-3px;height:19px;width:12px" loading="lazy">表示所有整数(正、负、零)ℤ {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="整数的加号次方" style="vertical-align:-3px;height:20px;width:22px" loading="lazy">表示正整数ℤ + {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
" class="Wirisformula" role="math" alt="直自然数" style="vertical-align:-3px;height:19px;width:14px" loading="lazy">表示自然数(0,1,2,3…)ℕ {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
分段函数是什么?
- 分段函数由不同的函数定义取决于输入所在的区间
- 如。
" class="Wirisformula" role="math" alt="空格f开括号x闭括号等于开花括号表行单元格x加1端单元格行单元格2 x减4端单元格行单元格x平方端单元格结束表空白表行单元格x小于等于5端单元格行单元格5小于x小于10端单元格行单元格10小于等于x小于等于20端单元格结束表关闭" style="vertical-align:-34px;height:81px;width:234px" loading="lazy">f x = x + 1 2 x - 4 x 2 x ≤ 5 5 < x < 10 10 ≤ x ≤ 20 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 如。
- 各个函数的区域不能重叠
- 求分段函数的某一特定值
" class="Wirisformula" role="math" alt="空间x = k" style="vertical-align:-6px;height:22px;width:42px" loading="lazy">x = k {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 找出哪个区间包含
" class="Wirisformula" role="math" alt="空间k" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">k {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 替代
" class="Wirisformula" role="math" alt="空间x = k" style="vertical-align:-6px;height:22px;width:42px" loading="lazy">转换成相应的函数x = k {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 找出哪个区间包含
- 这个函数可能是连续的,也可能不是在音程的末尾
- 在上面的例子中,函数是
- 连续在
" class="Wirisformula" role="math" alt="X = 5" style="vertical-align:-6px;height:22px;width:37px" loading="lazy">作为x = 5 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="5 + 1 = 2左括号5右括号- 4" style="vertical-align:-6px;height:22px;width:109px" loading="lazy">5 + 1 = 2 ( 5 ) - 4 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 不是连续的
" class="Wirisformula" role="math" alt="X = 10" style="vertical-align:-6px;height:22px;width:46px" loading="lazy">作为x = 10 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="2左括号10右括号- 4不等于10的平方" style="vertical-align:-6px;height:23px;width:108px" loading="lazy">2 ( 10 ) - 4 ≠ 10 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 连续在
- 在上面的例子中,函数是
考试技巧
- 问题可以指“最大可能的域”
- 这通常是
" class="Wirisformula" role="math" alt="X元素的实数" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">除非x ∈ ℝ {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="直自然数" style="vertical-align:-3px;height:19px;width:14px" loading="lazy">,ℕ {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="直整数" style="vertical-align:-3px;height:19px;width:12px" loading="lazy">或ℤ {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="直有理数" style="vertical-align:-3px;height:19px;width:16px" loading="lazy">已经说过了ℚ {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} - 通常也有例外
- 例如平方根;
" class="Wirisformula" role="math" alt="X大于等于0" style="vertical-align:-6px;height:22px;width:36px" loading="lazy">对于包含以下内容的函数x ≥ 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="根号x" style="vertical-align:-6px;height:26px;width:29px" loading="lazy">x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} - 例如互反函数;
" class="Wirisformula" role="math" alt="X不等于2" style="vertical-align:-6px;height:22px;width:37px" loading="lazy">对于有分母的函数x ≠ 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="左括号x - 2右括号" style="vertical-align:-6px;height:22px;width:49px" loading="lazy">( x - 2 ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"}
- 例如平方根;
- 这通常是
工作的例子
对于函数
一)
写下的值 f ( 7 ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f左括号7右括号" style="vertical-align:-6px;height:22px;width:35px" loading="lazy">.
b)
求的范围 f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f左括号x右括号" style="vertical-align:-6px;height:22px;width:36px" loading="lazy">.