绝对值函数求逆及定义域值域分析

GeoGebra 命令及参数

1. LaTeX(steps) → counter
2. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → Problem
3. LaTeX("\text{Write the inverse of the function and simplify it.}") → text4
4. If(a ≟ 1, LaTeX["f(x)"], If[a ≟ -1, LaTeX["-f(x)"], LaTeX[(a * "f(x)")]]) → first
5. If(b ≟ 0, "", If[b < 0, LaTeX[("-" * abs(b))], LaTeX[("+" * b)]]) → second
6. If(a ≟ 1, LaTeX["f^{-1}(x)"], If[a ≟ -1, LaTeX["-f^{-1}(x)"], LaTeX[(a * "f^{-1}(x)")]]) → firstb
7. LaTeX("\text{Solve for }\textcolor{#006633}{f^{-1}(x)}.") → text6
8. LaTeX("\text{Handle the absolute value by working with two cases.}") → text7
9. LaTeX("\text{Determine the domains and ranges of }\textcolor{#000099}{f(x)}\text{ and }\textcolor{#006633}{f^{-1}(x)}.") → text1
10. LaTeX("\text{State the final results.}") → text9
11. LaTeX("\text{Graph both functions and their inverses together.}") → text14
12. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text13
13. TableText({{LaTeX[(("\textcolor{#000099}{f(x)=" * f) * "}")], LaTeX["\text{Original function}"]}, {LaTeX[((("\textcolor{#006633}{x=\left|" * first) * second) * "\right|}")], LaTeX["\text{Switch the variables.}"]}, {LaTeX[((("\textcolor{#006633}{x=\left|" * firstb) * second) * "\right|}")], LaTeX["\text{Since this defines the inverse function, change to }\textcolor{#006633}{f^{-1}(x)}."]}}) → text5
14. LaTeX("\text{In order for a function to have an inverse, the function must be one-to-one.}\cr \text{Since absolute value functions are V-shaped, the domain of }\textcolor{#000099}{f(x)}\text{ needs to be restricted.}\cr \text{The range(s) of }\textcolor{#006633}{f^{-1}(x)}\text{ will be the restricted domain(s) of }\textcolor{#000099}{f(x)}.\cr \text{The domain of }\textcolor{#006633}{f^{-1}(x)}\text{ will be the range of }\textcolor{#000099}{f(x)}.") → text2
15. TableText({{LaTeX[""], LaTeX["\textcolor{#000099}{f(x)}"], LaTeX["\textcolor{#006633}{f^{-1}(x)}"]}, {LaTeX["\text{Domain:}"], LaTeX["\text{all real numbers;}"], LaTeX["\textcolor{#006633}{[0,+\infty)}"]}, {LaTeX[""], LaTeX[(("\text{split domain where }\textcolor{#000099}{" * Simplify[(a * x) + b]) * "=0}")], LaTeX["\textcolor{#006633}{}"]}, {LaTeX[""], LaTeX["\text{ to get restricted domains where }"], LaTeX["\textcolor{#006633}{}"]}, {LaTeX[""], LaTeX["\text{ }\textcolor{#000099}{f(x)}\text{ is one-to-one}"], LaTeX["\textcolor{#006633}{}"]}, {LaTeX[""], LaTeX[(((("\textcolor{#000099}{(-\infty," * Simplify[((-b)) / a]) * "]}\text{ and }\textcolor{#000099}{[") * Simplify[((-b)) / a]) * ",+\infty)}")], LaTeX["\textcolor{#006633}{}"]}, {LaTeX["\text{Range:}"], LaTeX["\text{The vertex of the function is at}"], LaTeX[(((("\textcolor{#006633}{(-\infty," * Simplify[((-b)) / a]) * "]}\text{ and }\textcolor{#006633}{[") * Simplify[((-b)) / a]) * ",+\infty)}")]}, {LaTeX[""], LaTeX["\text{ }\textcolor{#000099}{x=0}\text{ and the function opens up.}"], LaTeX[""]}, {LaTeX[""], LaTeX["\textcolor{#000099}{[0,+\infty)}"], LaTeX[""]}}) → text3
16. TableText({{LaTeX[((("\textcolor{#006633}{x=" * firstb) * second) * "}")], LaTeX["\text{Set up two cases.}"], LaTeX[((("\textcolor{#006633}{-x=" * firstb) * second) * "}")]}, {LaTeX[(((("\textcolor{#006633}{" * Simplify[x - b]) * "=") * firstb) * "}")], LaTeX[(("\text{Subtract }\textcolor{#006633}{" * b) * "}\text{ from both sides.}")], LaTeX[(((("\textcolor{#006633}{" * Simplify[(-x) - b]) * "=") * firstb) * "}")]}, {LaTeX[(((("\textcolor{#006633}{\frac{" * Simplify[x - b]) * "}{") * a) * "}=f^{-1}(x)}")], LaTeX[(("\text{Divide both sides by }\textcolor{#006633}{" * a) * "}.")], LaTeX[(((("\textcolor{#006633}{\frac{" * Simplify[(-x) - b]) * "}{") * a) * "}=f^{-1}(x)}")]}, {LaTeX[(("\textcolor{#006633}{f^{-1}(x)=" * g_1) * "}")], LaTeX["\text{Simplify and use the Symmetric Property of Equality.}"], LaTeX[(("\textcolor{#006633}{f^{-1}(x)=" * g_2) * "}")]}}) → text8
17. LaTeX((((((((((((("\text{On the restricted domain }\textcolor{#000099}{[" * Simplify[((-b)) / a]) * ",+\infty)}\text{, the inverse of }\textcolor{#000099}{f(x)=") * f) * "}\text{ is }\textcolor{#006633}{f^{-1}(x)=") * g_1) * "}\text{ where }\textcolor{#006633}{[0,+\infty)}\text{.}\cr \text{On the restricted domain }\textcolor{#000099}{(-\infty,") * Simplify[((-b)) / a]) * "]}\text{, the inverse of }\textcolor{#000099}{f(x)=") * f) * "}\text{ is }\textcolor{#006633}{f^{-1}(x)=") * g_2) * "}\text{ where }\textcolor{#006633}{[0,+\infty)}. ")) → text10
18. LaTeX((("\text{Graph the function on the restricted domain }\cr\textcolor{#000099}{[" * Simplify[((-b)) / a]) * ",+\infty)}\text{ and graph its inverse.}")) → text11
19. LaTeX((("\text{Graph the function on the restricted domain }\cr\textcolor{#000099}{(-\infty," * Simplify[((-b)) / a]) * "]}\text{ and graph its inverse.}")) → text12