函数复合运算示例

几何画板及绘图内容加载中，请稍等……

2014-07-29 03:21

所属专题 CCGPS AA Unit 6 Lesson 7

本示例演示如何计算函数f(x) = 2x³ - 1和g(x) = (x+1/2)^(1/3)的复合函数f(g(x))和g(f(x))，并验证它们是否互为反函数。涉及的核心数学概念包括函数、反函数和复合函数。使用了LaTeX、If、Max等GeoGebra命令进行构造。适合高中数学教学和学习。

GeoGebra 命令及参数

LaTeX(steps) → counter
LaTeX("\text{Move the slider to solve the problem.}") → text0
Corner(1) → A
Corner(3) → B
LaTeX("\text{}") → Problem
Polynomial((a * x^(c)) + b) → f
If(IsInteger[c / 2], LaTeX["\pm"], "") → evenodd
LaTeX("\textcolor{#CC0000}{f(g(x))=x}") → text10
LaTeX("\textcolor{#CC0000}{g(f(x))=x}") → text21
Max(Zip[If[Defined[x(a)], abs(x(a)), 555], a, list1] / {555}) → maxzeros
LaTeX((((((((((("\textcolor{#CC0000}{f(g(x))=" * a) * "\left(") * evenodd) * "") * g) * "\right)^{") * c) * "}+(") * b) * ")}")) → text6
LaTeX("\text{Simplify the powers.}") → text7
LaTeX("\text{Simplify the powers.}") → text20
LaTeX((((((("\text{Find }\textcolor{#CC0000}{f(g(x))}\text{ using }\textcolor{#000099}{f(x)=" * f) * "}\text{ and }\textcolor{#006633}{g(x) = ") * evenodd) * "") * g) * "}.")) → text1
LaTeX("\text{Original function }\textcolor{#000099}{f(x)}") → text3
LaTeX((("\text{Substitute }\textcolor{#006633}{" * g) * "}\text{ for }\textcolor{#000099}{x}\text{ and }\cr\text{name the new function }\textcolor{#CC0000}{f(g(x))}.")) → text5
LaTeX("\text{Simplify.}") → text9
LaTeX((((((("\text{Therefore }\textcolor{#CC0000}{f(g(x))=x}\text{ when }\textcolor{#000099}{f(x)= " * f) * "}\text{ and }\textcolor{#006633}{g(x)=") * evenodd) * "") * g) * "}.")) → text11
LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → text4
LaTeX((((((("\text{Find }\textcolor{#CC0000}{g(f(x))}\text{ using }\textcolor{#000099}{f(x)=" * f) * "}\text{ and }\textcolor{#006633}{g(x) = ") * evenodd) * "") * g) * "}.")) → text12
LaTeX("\text{Original function }\textcolor{#006633}{g(x)}") → text14
LaTeX("\text{We have determined that }\textcolor{#CC0000}{f(g(x)) = x}\text{ and }\textcolor{#CC0000}{g(f(x)) = x}\text{. }\cr\text{By substitution, }\textcolor{#CC0000}{f(g(x)) = g(f(x))};\text{ therefore, }\textcolor{#006633}{g(x)}\text{ is the inverse of }\textcolor{#000099}{f(x)}.") → text24
LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
LaTeX("\text{State a conclusion about whether }\textcolor{#006633}{g(x)}\text{ is the inverse of }\textcolor{#000099}{f(x)}\text{ based on the }\cr\text{results for }\textcolor{#CC0000}{f(g(x))}\text{ and }\textcolor{#CC0000}{g(f(x))}.") → text23
LaTeX((("\text{The notation }\textcolor{#CC0000}{f(g(x))}\text{ means that the input for }\textcolor{#000099}{f(x)}\text{ is equal to the output of }\textcolor{#006633}{g(x)}\text{.}\cr\text{In other words, to find }\textcolor{#CC0000}{f(g(x))}\text{, replace }\textcolor{#000099}{x}\text{ in the value of the function }\textcolor{#000099}{f(x)}\text{ with the }\cr\text{value of the function }\textcolor{#006633}{g(x)}\text{. Therefore, replace }\textcolor{#000099}{x}\text{ with }\textcolor{#006633}{" * g) * "}\text{ and then}\cr\text{solve the resulting composition of functions.}")) → text2
LaTeX((((((((("\textcolor{#CC0000}{f(g(x))=" * a) * "\left(\frac{") * Simplify[x - b]) * "}{") * a) * "}\right)+(") * b) * ")}")) → text8
LaTeX((("\text{The notation }\textcolor{#CC0000}{g(f(x))}\text{ means that the input for }\textcolor{#006633}{g(x)}\text{ is equal to the output of }\textcolor{#000099}{f(x)}\text{. }\cr\text{In other words, to find }\textcolor{#CC0000}{g(f(x))}\text{, replace }\textcolor{#006633}{x}\text{ in the value of the function }\textcolor{#006633}{g(x)}\text{ with }\cr\text{the value of the function }\textcolor{#000099}{f(x)}\text{. Therefore, replace }\textcolor{#006633}{x}\text{ with }\textcolor{#000099}{" * f) * "}\text{ and then solve}\cr\text{the resulting composition of functions.}")) → text13
LaTeX((("\textcolor{#006633}{g(x)=" * g) * "}")) → text15
LaTeX((("\text{Substitute }\textcolor{#000099}{" * f) * "}\text{ for }\textcolor{#006633}{x}\text{ and }\cr\text{name the new function }\textcolor{#CC0000}{g(f(x))}.")) → text16
LaTeX((((((((("\textcolor{#CC0000}{g(f(x))=\left(\frac{(" * f) * ")-(") * b) * ")}{") * a) * "}\right)^{1/") * c) * "}}")) → text17
LaTeX((((((("\textcolor{#CC0000}{g(f(x))=" * evenodd) * "\left(x^") * c) * "\right)^{1/") * c) * "}}")) → text19
LaTeX("\text{Simplify.}") → text18
LaTeX((((((("\text{}\textcolor{#CC0000}{g(f(x))=x}\text{ when }\textcolor{#000099}{f(x)= " * f) * "}\text{ and }\textcolor{#006633}{g(x)=") * evenodd) * "") * g) * "}.")) → text22

AI 绘图提示语参考

绘制函数 f(x) = 2x³ - 1 和 g(x) = (x+1/2)^(1/3) 的图像，并计算它们的复合函数 f(g(x)) 和 g(f(x))，验证它们是否互为反函数。首先绘制 f(x) 和 g(x)，然后计算 f(g(x)) 和 g(f(x))，简化表达式并验证结果是否等于 x。

以上提示语为根据原始绘图命令整理而来，仅供参考。