验证函数 f(x)=±√(x+3) 与 g(x)=x²-3 互为反函数

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2014-07-29 03:09

函数变换函数图像

所属专题 CCGPS AA Unit 6 Lesson 7

① 该范例演示如何通过复合函数 f(g(x)) 与 g(f(x)) 验证两个函数互为反函数的图形与代数关系。② 涉及核心数学概念：函数复合、反函数定义、定义域与值域的对应、平方根函数的正负分支。③ 作图要点：使用了 LaTeX、Corner、If、Simplify、Max 等命令，配合 stepMax、stepper、steps、counter、text0、Original、Randomize、ZoomFit、button1、A、B、txmin、txmax、tymin、tymax、Problem、a、b、c、f、evenodd、f2、g、d、text4、text6、text8、text9、text10、text21、text23、text24、list1、maxzeros、range、xmax、xmin、ymax、ymin、text3 等对象实现动态展示与步骤控制。④ 适合高中数学“函数与导数”章节或大学预科课程，作为反函数验证的教学演示与自学素材。

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

AI 绘图提示语参考

绘制函数 f(x) = ±√(x+3) 和 g(x) = x²-3 的图形，并验证它们是否互为反函数。首先绘制 f(x) 的正负两个分支，然后绘制 g(x)。接着计算 f(g(x)) 和 g(f(x))，简化表达式，观察结果是否等于 x，以验证它们是否互为反函数。

以上提示语为根据原始绘图命令整理而来，仅供参考。