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求函数f(x)=4x²的反函数及其定义域

求函数f(x)=4x²的反函数及其定义域
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2014-07-29 22:12
函数解析几何微积分
所属专题 CCGPS AA Unit 6 Lesson 7
本例通过求解函数f(x)=4x²的反函数g(x)，并确定其定义域。涉及的核心数学概念包括反函数和函数的定义域。使用了GeoGebra中的LaTeX、If、TableText等命令进行构造。适合高中数学教学和学习。

GeoGebra 命令及参数

  1. LaTeX(steps) → counter
  2. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → Problem
  3. LaTeX("\text{Switch the domain and function variables, and then rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{g(x).}") → text1
  4. LaTeX((("\text{Since the inverse function results from switching the input and output variables}\cr\text{of the original function, identify which variable represents the input and which}\cr\text{represents the output. For the function }\textcolor{#000099}{f(x) = " * f) * "}\text{, }\textcolor{#000099}{x}\text{ is the input (independent)}\cr\text{variable, and }\textcolor{#000099}{f(x)}\text{ is the output (dependent) variable. Therefore, swap }\textcolor{#000099}{f(x)}\text{ and }\textcolor{#000099}{x}.")) → text2
  5. LaTeX("\text{Next, rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{g(x)}\text{, since this new relation represents the inverse of the}\cr\text{original function.}") → text4
  6. LaTeX((((("\text{The resulting inverse relation, }\textcolor{#006633}{x = " * a) * "[g(x)]^2}\text{, is the possible inverse function of}\cr\text{the original function. It must be evaluated to determine whether it has }\cr\text{one-to-one correspondence with the original function, }\textcolor{#000099}{f(x) = ") * a) * "x^2}.")) → text6
  7. LaTeX("\text{Solve the possible inverse for }\textcolor{#006633}{g(x)}.") → text7
  8. LaTeX("\text{Determine the domain of }\textcolor{#006633}{g(x)}.") → text9
  9. LaTeX("\text{Determine the range of }\textcolor{#006633}{g(x)}.") → text13
  10. LaTeX("\text{We need to compare the range of }\textcolor{#006633}{g(x)}\text{ to the domain of }\textcolor{#000099}{f(x)}.") → text14
  11. LaTeX("\text{Determine the domain of }\textcolor{#000099}{f(x)}.") → text16
  12. LaTeX("\text{Determine whether the function }\textcolor{#000099}{f(x)}\text{ exhibits one-to-one correspondence.}") → text18
  13. LaTeX("\text{Determine the parts of its domain over which }\textcolor{#000099}{f(x)}\text{ exhibits one-to-one}\cr\text{correspondence.}") → text21
  14. LaTeX("\text{Determine the range of }\textcolor{#000099}{f(x)}.") → text23
  15. LaTeX("\text{Match domains to ranges to find the inverse(s) of }\textcolor{#000099}{f(x)}.") → text25
  16. LaTeX("\text{Check that the range of }\textcolor{#000099}{f(x)}\text{ matches the domain of }\textcolor{#006633}{g(x)}.") → text28
  17. If(a ≥ 0, "", "-") → signtext
  18. If(a ≥ 0, "non-negative", "negative") → numberwords
  19. If(a ≥ 0, LaTeX["[0, + \infty)"], LaTeX["(-\infty,0]"]) → domaing
  20. LaTeX((("\text{Because }\textcolor{#000099}{f(x)}\text{ is a quadratic function and }\textcolor{#000099}{x^2}\text{ can never be negative, the}\cr\text{range of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{" * domaing) * "}.")) → text24
  21. If(a ≥ 0, sqrt(x) / asqrt, sqrt((-x)) / asqrt) → g_1
  22. If(a ≥ 0, ((-sqrt(x))) / asqrt, ((-sqrt((-x)))) / asqrt) → g_2
  23. LaTeX((((("\text{The range of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{" * domaing) * "}\text{. The domain of }\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{") * domaing) * "}.")) → text29
  24. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text31
  25. TableText({{LaTeX[(("\textcolor{#000099}{f(x)=" * a) * "x^2}")], LaTeX["\text{Original function}"]}, {LaTeX[(("\textcolor{#000099}{x=" * a) * "[f(x)]^2}")], LaTeX["\text{Switch }\textcolor{#000099}{f(x)}\text{ and }\textcolor{#000099}{x}."]}}) → text3
  26. TableText({{LaTeX[(("\textcolor{#000099}{x=" * a) * "[f(x)]^2}")], LaTeX["\text{Equation with variables switched}"]}, {LaTeX[(("\textcolor{#006633}{x=" * a) * "[g(x)]^2}")], LaTeX["\text{Rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{g(x)}."]}}) → text5
  27. LaTeX("\text{The function }\textcolor{#000099}{f(x)}\text{ can only have an inverse over the part(s) of its domain}\cr\text{for which it exhibits one-to-one correspondence.}") → text19
  28. LaTeX((((((((("\text{We can determine that the function }\textcolor{#000099}{f(x) = " * a) * "x^2}\text{ does not show}\cr\text{one-to-one correspondence for all domain values because }\textcolor{#000099}{f(a) = ") * a) * "a^2}\text{ and }\cr\textcolor{#000099}{f(–a) = ") * a) * "(–a)^2 = ") * a) * "a^2}\text{. Since }\textcolor{#000099}{f(x)}\text{ has the same value for two different}\cr\text{values of }\textcolor{#000099}{x}\text{, }\textcolor{#000099}{f(x)}\text{ is not one-to-one.}\cr\text{An equivalant graphical test is the horizontal line test. If any horizontal}\cr\text{line touches the graph of }\textcolor{#000099}{f(x)}\text{ more than once, then }\textcolor{#000099}{f(x)}\text{ is not one-to-one.}")) → text20
  29. LaTeX("\text{We need to know where }\textcolor{#006633}{g(x)}\text{ is defined in order to know its domain,}\cr\text{which we want to compare to the range of }\textcolor{#000099}{f(x)}.") → text10
  30. LaTeX((((("\text{The function }\textcolor{#006633}{g(x)}\text{ is a radical function; specifically, it is a square root} \cr\text{function. Recall that the value of the expression inside the square root} \cr\text{must be positive in order for the function to be defined. The expression} \cr\text{under the square root is }\textcolor{#006633}{" * signtext) * "x}\text{; therefore, the values of }\textcolor{#006633}{x}\text{ are limited to}\cr\text{") * numberwords) * " numbers.}")) → text11
  31. LaTeX("\text{There are no limitations on the output values; therefore, the range}\cr\text{of }\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{(−\infty, +\infty)}.") → text15
  32. LaTeX("\text{There are no limitations on the input values; therefore, the domain}\cr\text{of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{(−\infty, +\infty)}.") → text17
  33. LaTeX("\text{ The restricted domains of }\textcolor{#000099}{f(x)}\text{, on which }\textcolor{#000099}{f(x)}\text{ is one-to-one, are }\textcolor{#000099}{(−\infty, 0]}\cr\text{ and }\textcolor{#000099}{[0, +\infty)}\text{. Only the point }\textcolor{#000099}{(0, 0)}\text{ is not restricted, because all of}\cr\text{ the other points do not have one-to-one correspondence. The horizontal}\cr\text{ line crosses the graph at two points for all other points on the graph.}") → text22
  34. LaTeX("\text{In order for a function to have an inverse, it must be one-to-one. Therefore, }\cr\text{match the restricted domains of }\textcolor{#000099}{f(x)}\text{ to the range of }\textcolor{#006633}{g(x)}.") → text26
  35. TableText({{LaTeX["\text{Restricted domain of }\textcolor{#000099}{f(x)}\text{:}"], LaTeX["\text{Inverse function with matching range:}"]}, {LaTeX["\textcolor{#000099}{[0,+\infty)}"], LaTeX[(("\textcolor{#006633}{g(x)=" * g_1) * "}")]}, {LaTeX["\textcolor{#000099}{(-\infty,0]}"], LaTeX[(("\textcolor{#006633}{g(x)=" * g_2) * "}")]}}) → text27
  36. LaTeX((((((((("\text{The inverse of }\textcolor{#000099}{f(x)=" * a) * "x^2}\text{ on the restricted domain }\textcolor{#000099}{0≤x<+\infty}\text{ is }\textcolor{#006633}{g(x)=") * g_1) * "}.\cr \text{The inverse of }\textcolor{#000099}{f(x)=") * a) * "x^2}\text{ on the restricted domain }\textcolor{#000099}{-\infty<x≤0}\text{ is }\textcolor{#006633}{g(x)=") * g_2) * "}.")) → text30
  37. LaTeX((("\text{Thus, the domain of }\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{" * domaing) * "}\text{ because }\textcolor{#006633}{x}\text{ is raised to the }\textcolor{#006633}{\frac{1}{2}}\text{ power}\cr\text{ (recall that }\textcolor{#006633}{\sqrt{x} = x^\frac{1}{2}}\text{).}")) → text12
  38. TableText({{LaTeX[(("\textcolor{#006633}{x=" * a) * "[g(x)]^2}")], LaTeX["\text{Possible inverse}"]}, {LaTeX[(("\textcolor{#006633}{\frac{x}{" * a) * "}=[g(x)]^2}")], LaTeX[(("\text{Divide both sides by }\textcolor{#006633}{" * a) * "}.")]}, {LaTeX[(("\textcolor{#006633}{[g(x)]^2}=\frac{x}{" * a) * "}")], LaTeX["\text{Use Symmetric Property of Equality.}"]}, {LaTeX[(((("\textcolor{#006633}{g(x)}=\pm\sqrt{\frac{" * signtext) * "x}{") * abs(a)) * "}}")], LaTeX["\text{Find both roots of the equation. (Recall }(-1)^2=(1)^2\text{.)}"]}, {LaTeX[(((("\textcolor{#006633}{g(x)}=\pm\frac{\sqrt{" * signtext) * "x}}{\sqrt{") * abs(a)) * "}}")], LaTeX["\text{Distribute the square root operation.}"]}, {LaTeX[(((("\textcolor{#006633}{g(x)}=\pm\frac{\sqrt{" * signtext) * "x}}{") * asqrt) * "}")], LaTeX["\text{Simplify.}"]}, {LaTeX[(((("\textcolor{#006633}{g(x)}=\pm\frac{1}{" * asqrt) * "}\sqrt{") * signtext) * "x}")], LaTeX["\text{Separate out the coefficient.}"]}}) → text8

AI 绘图提示语参考

画出函数 f(x) = 4x² 的图像，并找到它的反函数 g(x)。首先，交换 x 和 f(x) 的位置，得到 x = 4[g(x)]²。然后，将 f(x) 重命名为 g(x)，得到可能的反函数。接着，解方程找到 g(x) 的具体形式，考虑根号内的值必须非负，得到 g(x) = ±(1/2√4)x^(1/2)。最后，确定 g(x) 的定义域为 (-∞, 0] 和 [0, +∞)，并检查 f(x) 的定义域和 g(x) 的值域是否匹配。
以上提示语为根据原始绘图命令整理而来，仅供参考。