Inverse of an Absolute Value Function and Domain/Range Analysis

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2014-08-06 01:41

In collections UCSS Secondary Math III Unit 4A Lesson 1

This example demonstrates how to find the inverse of the absolute value function f(x) = |x + 2| and analyze its domain and range. Key math concepts include absolute value functions, inverse functions, domain, and range. GeoGebra commands and objects used are LaTeX for mathematical expressions, If for conditional statements, TableText for creating text tables, stepMax, stepper, steps, counter for step-by-step demonstration, and objects like f, g_1, g_2 for function graphs, mirror for reflection transformation. Suitable for high school students learning function properties and inverse function concepts.

GeoGebra Commands & Parameters

LaTeX(steps) → counter
LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → Problem
LaTeX("\text{Write the inverse of the function and simplify it.}") → text4
If(a ≟ 1, LaTeX["f(x)"], If[a ≟ -1, LaTeX["-f(x)"], LaTeX[(a * "f(x)")]]) → first
If(b ≟ 0, "", If[b < 0, LaTeX[("-" * abs(b))], LaTeX[("+" * b)]]) → second
If(a ≟ 1, LaTeX["f^{-1}(x)"], If[a ≟ -1, LaTeX["-f^{-1}(x)"], LaTeX[(a * "f^{-1}(x)")]]) → firstb
LaTeX("\text{Solve for }\textcolor{#006633}{f^{-1}(x)}.") → text6
LaTeX("\text{Handle the absolute value by working with two cases.}") → text7
LaTeX("\text{Determine the domains and ranges of }\textcolor{#000099}{f(x)}\text{ and }\textcolor{#006633}{f^{-1}(x)}.") → text1
LaTeX("\text{State the final results.}") → text9
LaTeX("\text{Graph both functions and their inverses together.}") → text14
LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text13
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
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
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
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
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
LaTeX((("\text{Graph the function on the restricted domain }\cr\textcolor{#000099}{[" * Simplify[((-b)) / a]) * ",+\infty)}\text{ and graph its inverse.}")) → text11
LaTeX((("\text{Graph the function on the restricted domain }\cr\textcolor{#000099}{(-\infty," * Simplify[((-b)) / a]) * "]}\text{ and graph its inverse.}")) → text12

AI Drawing Prompt Reference

Draw the graph of the absolute value function f(x) = |x+2| and its inverse, analyzing their domains and ranges. First, plot the V-shaped graph of the absolute value function and determine its domain and range. Then, find the inverse function and plot its graph on two intervals. Finally, draw both the original function and its inverse on the same coordinate system, labeling their domains and ranges.

The prompt above was compiled from the original drawing commands and is for reference only.