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求函数反函数及定义域示例

求函数反函数及定义域示例
几何画板及绘图内容加载中，请稍等……
2014-08-06 01:37
函数解析几何函数图像
所属专题 UCSS Secondary Math III Unit 4A Lesson 1
本范例演示如何求函数f(x) = x(x-1)的反函数f⁻¹(x)并确定其定义域。涉及的核心数学概念包括反函数的定义、求反函数的方法以及函数定义域的确定。作图要点使用了GeoGebra的LaTeX、If、Simplify、TableText等命令和对象，如f、text0、text1等，通过这些命令和对象实现了反函数的求解和展示。适合高中数学教学和学生学习反函数相关知识。

GeoGebra 命令及参数

  1. LaTeX(steps) → counter
  2. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → Problem
  3. 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
  4. LaTeX("\text{Determine the domain of }\textcolor{#000099}{f(x)}.") → text16
  5. LaTeX("\text{Determine whether the function }\textcolor{#000099}{f(x)}\text{ exhibits one-to-one correspondence.}") → text18
  6. LaTeX("\text{Determine the range of }\textcolor{#000099}{f(x)}.") → text23
  7. LaTeX("\text{Match domains to ranges to find the inverse(s) of }\textcolor{#000099}{f(x)}.") → text25
  8. If(a ≥ 0, "", "-") → signtext
  9. If(a ≥ 0, "non-negative", "negative") → numberwords
  10. If(a ≥ 0, LaTeX["[0, + \infty)"], LaTeX["(-\infty,0]"]) → domaing
  11. LaTeX("\text{Switch the domain and function variables, and then rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{f^{-1}(x).}") → text1
  12. LaTeX("\text{Next, rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{f^{-1}(x)}\text{, since this new relation represents the inverse of the}\cr\text{original function.}") → text4
  13. Simplify(ac / ((2 * ab))) → gcon
  14. If(a ≟ 1, LaTeX["f^{-1}(x)"], If[a ≟ -1, LaTeX["-f^{-1}(x)"], LaTeX[(a * "f^{-1}(x)")]]) → inversefront
  15. If(b ≟ 1, LaTeX["f^{-1}(x)"], If[b ≟ -1, LaTeX["-f^{-1}(x)"], LaTeX[(b * "f^{-1}(x)")]]) → inversemid
  16. If(c < 0, LaTeX[("+" * abs(c))], LaTeX[("-" * c)]) → inverseend
  17. LaTeX(((((inversefront * "(") * inversemid) * inverseend) * ")")) → inversefunc
  18. LaTeX((((("\text{The resulting inverse relation, }\textcolor{#006633}{x = " * inversefunc) * "}\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) = ") * f) * "}.")) → text6
  19. If(a ≟ 1, LaTeX["f(x)"], If[a ≟ -1, LaTeX["-f(x)"], LaTeX[(a * "f(x)")]]) → inversefronta
  20. If(b ≟ 1, LaTeX["f(x)"], If[b ≟ -1, LaTeX["-f(x)"], LaTeX[(b * "f(x)")]]) → inversemida
  21. LaTeX(((((inversefronta * "(") * inversemida) * inverseend) * ")")) → inversefunca
  22. LaTeX("\text{Solve the possible inverse for }\textcolor{#006633}{f^{-1}(x)}.") → text7
  23. If(ab ≟ -1, LaTeX["-[f^{-1}(x)]^2"], If[ab ≟ 1, LaTeX["[f^{-1}(x)]^2"], LaTeX[(ab * "[f^{-1}(x)]^2")]]) → quadfront
  24. If(ac ≟ -1, LaTeX["+f^{-1}(x)"], If[ac ≟ 1, LaTeX["-f^{-1}(x)"], If[ac < 0, LaTeX[(("+" * abs(ac)) * "f^{-1}(x)")], LaTeX[(("-" * ac) * "f^{-1}(x)")]]]) → quadmid
  25. LaTeX("\text{Determine the domain of }\textcolor{#006633}{f^{-1}(x)}.") → text9
  26. LaTeX("\text{Determine the range of }\textcolor{#006633}{f^{-1}(x)}.") → text13
  27. LaTeX("\text{We need to compare the range of }\textcolor{#006633}{f^{-1}(x)}\text{ to the domain of }\textcolor{#000099}{f(x)}.") → text14
  28. Simplify(f) → fsimp
  29. LaTeX("\text{The function }\textcolor{#000099}{f(x)}\text{ can only have an inverse over the parts of its domain where it exhibits}\cr\text{one-to-one correspondence.}") → text19
  30. LaTeX((((("\text{We can determine that the function }\textcolor{#000099}{f(x) =" * f) * "}\text{ does not show one-to-one}\cr\text{correspondence for all domain values by looking at it in standard form: }\textcolor{#000099}{f(a) = ") * fsimp) * "}.\cr\text{We can now more easily see that }\textcolor{#000099}{f(x)}\text{ is a quadratic function, and that its graph is a parabola.}")) → text20
  31. LaTeX("\text{Determine the parts of its domain over which }\textcolor{#000099}{f(x)}\text{ exhibits one-to-one correspondence.}") → text21
  32. LaTeX("\text{Because, }\textcolor{#000099}{f(x)}\text{ is quadratic, the parts of the graph of }\textcolor{#000099}{f(x)}\text{ that show a one-to-one correspondence}\cr\text{lie on either side of its axis of symmetry. This axis divides the graph of the function into two}\cr\text{halves. If }\textcolor{#000099}{f(x)}\text{ is written as }\textcolor{#000099}{ax^2 + bx + c}\text{, the equation of its axis of symmetry is given by }\cr\textcolor{#000099}{x = −\frac{b}{2a}}.\text{ The domains of the halves of the graph of }\textcolor{#000099}{f(x)}\text{ are defined by this value of }\textcolor{#000099}{x}.") → text22
  33. LaTeX((((("\textcolor{#000099}{x = −\frac{b}{2a}=\frac{" * ac) * "}{2(ab)}=") * Simplify[ac / ((2 * ab))]) * "}")) → text22a
  34. TableText({{LaTeX[(("\textcolor{#000099}{f(x)=" * f) * "}")], LaTeX["\text{Original function}"]}, {LaTeX[(("\textcolor{#000099}{x=" * inversefunca) * "}")], LaTeX["\text{Switch }\textcolor{#000099}{f(x)}\text{ and }\textcolor{#000099}{x}."]}}) → text3
  35. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text31
  36. LaTeX((("\text{In other words, because of the square root, the domain of }\textcolor{#006633}{f^{-1}(x)} \text{ is }\textcolor{#006633}{[" * Simplify[((-ac^(2))) / ((4 * b))]) * ",+∞)}.")) → text12
  37. LaTeX((((("\text{There are two restricted domains where }\textcolor{#000099}{f(x)}\text{ is one-to-one: }\textcolor{#000099}{(-\infty," * Simplify[ac / ((2 * ab))]) * "]}\text{ and }\textcolor{#000099}{[") * Simplify[ac / ((2 * ab))]) * ",+\infty)}.")) → text22b
  38. LaTeX((((("\text{The range of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{[" * Simplify[((-((a * c^(2))))) / ((4 * b))]) * ",+\infty)}\text{. The domain of }\textcolor{#006633}{f^{-1}(x)}\text{ is }\textcolor{#006633}{[") * Simplify[((-((a * c^(2))))) / ((4 * b))]) * ",+\infty)}.")) → text28
  39. LaTeX((((((("\text{The inverse of }\textcolor{#000099}{f(x)=" * f) * "}\text{ on the restricted domain }\textcolor{#000099}{[") * Simplify[ac / ((2 * ab))]) * ",+\infty)}\text{, is }\textcolor{#006633}{f^{-1}(x)=") * g_1) * "}.")) → text29
  40. LaTeX((((((("\text{The inverse of }\textcolor{#000099}{f(x)=" * f) * "}\text{ on the restricted domain }\textcolor{#000099}{(-\infty,") * Simplify[ac / ((2 * ab))]) * "]}\text{, is }\textcolor{#006633}{f^{-1}(x)=") * g_2) * "}.")) → text30
  41. TableText({{LaTeX[(("\textcolor{#006633}{x=" * inversefunc) * "}")], LaTeX["\text{Possible inverse}"]}, {LaTeX[((("\textcolor{#006633}{x=" * quadfront) * quadmid) * "}")], LaTeX["\text{Multiply out the right side.}"]}, {LaTeX[((("\textcolor{#006633}{0=" * quadfront) * quadmid) * "-x}")], LaTeX["\text{Subtract }\textcolor{#006633}{x}\text{ from both sides.}"]}, {LaTeX[(((((((("\textcolor{#006633}{f^{-1}(x)=\frac{" * ac) * "\pm\sqrt{(") * ac) * ")^2-4(") * ab) * ")(-x)}}{2(") * ab) * ")}}")], LaTeX["\text{Apply the quadratic formula. Treat }\textcolor{#006633}{x}\text{ as a constant and }\textcolor{#006633}{f^{-1}(x)}\text{ as the variable.}"]}, {LaTeX[(((("\textcolor{#006633}{f^{-1}(x)=" * gcon) * "\pm") * gfunc) * "}")], LaTeX["\text{Simplify.}"]}}) → text8
  42. LaTeX("\text{We need to know where }\textcolor{#006633}{f^{-1}(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
  43. LaTeX((((("\text{The function }\textcolor{#006633}{f^{-1}(x)}\text{ is a radical function; specifically, it is a square root function.} \cr\text{Recall that the value of the expression inside the square root must be positive}\cr\text{in order for the function to be defined. The expression under the square root is }\textcolor{#006633}{" * Simplify[acsq + ((4 * ab) * x)]) * "};\cr\text{therefore, the values of }\textcolor{#006633}{x}\text{ are limited to }\textcolor{#006633}{x\ge ") * Simplify[((-ac^(2))) / ((4 * b))]) * "}.")) → text11
  44. LaTeX((((((((("\text{There are no limitations on the output values; therefore, the range of }\textcolor{#006633}{f^{-1}(x)}\text{ is }\cr\textcolor{#006633}{(−\infty, +\infty)}\text{. However, }\textcolor{#006633}{f^{-1}(x)}\text{ is made of two pieces, and each piece has some}\cr\text{limitation on its range. The range of }\textcolor{#006633}{" * g_1) * "}\text{ is }\textcolor{#006633}{[") * Simplify[ac / ((2 * ab))]) * ",+∞)}\text{. The range of }\cr\textcolor{#006633}{") * g_2) * "}\text{ is }\textcolor{#006633}{(-∞,") * Simplify[ac / ((2 * ab))]) * "]}.")) → text15
  45. LaTeX("\text{There are no limitations on the input values; therefore, the domain }\text{of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{(−\infty, +\infty)}.") → text17
  46. TableText({{LaTeX[(("\textcolor{#000099}{x=" * inversefunca) * "}")], LaTeX["\text{Equation with variables switched}"]}, {LaTeX[(("\textcolor{#006633}{x=" * inversefunc) * "}")], LaTeX["\text{Rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{f^{-1}(x)}."]}}) → text5
  47. TableText({{LaTeX["\text{Restricted domain of }\textcolor{#000099}{f(x)}\text{:}"], LaTeX["\text{Inverse function with matching range:}"]}, {LaTeX[(("\textcolor{#000099}{[" * Simplify[ac / ((2 * ab))]) * ",+\infty)}")], LaTeX[(("\textcolor{#006633}{f^{-1}(x)=" * g_1) * "}")]}, {LaTeX[(("\textcolor{#000099}{(-\infty," * Simplify[ac / ((2 * ab))]) * "]}")], LaTeX[(("\textcolor{#006633}{f^{-1}(x)=" * g_2) * "}")]}}) → text26
  48. LaTeX("\text{Check that the range of }\textcolor{#000099}{f(x)}\text{ matches the domain of }\textcolor{#006633}{f^{-1}(x)}.") → text27
  49. LaTeX((((((("\text{Because }\textcolor{#000099}{f(x)}\text{ is a parabolic function, its range is limited. Since the coefficient of the quadratic term is positive, }\cr\text{the parabola opens up and the minimum of the range is the y-coordinate of the minimum of the parabola.}\cr\text{The x-coordinate of the minimum is known; it is the axis of symmetry, }\textcolor{#000099}{x=" * Simplify[ac / ((2 * ab))]) * "}\text{. To determine the y-coordinate,}\cr\text{substitute the equation of the axis of symmetry into the original function, }\textcolor{#000099}{f(x).}\cr\text{The result is }\textcolor{#000099}{y=") * Simplify[((-((a * c^(2))))) / ((4 * b))]) * "}\text{. Therefore, the range of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{[") * Simplify[((-((a * c^(2))))) / ((4 * b))]) * ",+\infty)}.")) → text24

AI 绘图提示语参考

画一个函数 f(x) = x(x-1) 的图像，并展示其反函数的求解过程。首先，绘制 f(x) 的图像，并标注其表达式。接着，通过交换变量并求解，得到反函数的表达式，并标注其图像。同时，分析 f(x) 的定义域和值域，确定反函数的定义域和值域，并用不同颜色区分原始函数和反函数。最后，标注出 f(x) 的单调区间，以及反函数在不同单调区间上的表达式。
以上提示语为根据原始绘图命令整理而来，仅供参考。