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10. REINFORCE Consider the function f(x) = -2x² +18. Fill in the following tables to describe f(x) and its inverse, f¹(x). Sketch a graph of f(x) and f(x) on the same graph grid. Is the inverse relation a function? Explain. f-¹(x) X f(x) agile Mind X -20 15 10 0 10 15 Page 5 of 8 20 Copyright Agile Mind, Inc. ® Content copyright Charles A. Dana Center, The University of Texas at Austin

10. REINFORCE Consider the function f(x) = -2x² +18. Fill in the following tables to describe f(x) and its inverse, f¹(x). Sketch a graph of f(x) and f(x) on the same graph grid. Is the inverse relation a function? Explain. f-¹(x) X f(x) agile Mind X -20 15 10 0 10 15 Page 5 of 8 20 Copyright Agile Mind, Inc. ® Content copyright Charles A. Dana Center, The University of Texas at Austin

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Educational Resource: Exploring Quadratic Functions and Their Inverses** --- **Consider the Function \( f(x) = -2x^2 + 18 \):** This exercise involves analyzing the quadratic function \( f(x) = -2x^2 + 18 \), filling in values for \( f(x) \) and its inverse \( f^{-1}(x) \), and then sketching both on a coordinate grid. It also includes determining if the inverse relation is a function. **Tables for Values:** - **\( x \) and \( f(x) \) Table:** - This table is for recording outputs \( f(x) \) for given inputs \( x \). - **\( x \) and \( f^{-1}(x) \) Table:** - This table is meant to record values of \( x \) with corresponding \( f^{-1}(x) \), the inverse relation. **Graph Details:** - The graph grid has axes ranging from -5 to 20 on the x-axis and -5 to 20 on the y-axis. - Students are expected to plot the quadratic function \( f(x) = -2x^2 + 18 \) and analyze its graph. - Additionally, plot the values for the inverse and explore the symmetry, particularly focusing on line \( y=x \) which can help visualize relation inverses. **Key Concepts:** - **Inverse Relation of a Function:** - For a quadratic like \( f(x) = -2x^2 + 18 \), the inverse may not be a function unless restricted on a specific interval because quadratics have a parabolic shape, which doesn’t pass the Horizontal Line Test. - **Determining if Inverse is a Function:** - Exploring if for each output there’s only one corresponding input in the inverse will be key. Graphically, the inverse would need to pass the Vertical Line Test if viewed as a function. --- This exercise assists in understanding function inverses, graphical representation, and conceptualizing whether certain mathematical inverses satisfy the criteria to be functions themselves.
Transcribed Image Text:**Educational Resource: Exploring Quadratic Functions and Their Inverses** --- **Consider the Function \( f(x) = -2x^2 + 18 \):** This exercise involves analyzing the quadratic function \( f(x) = -2x^2 + 18 \), filling in values for \( f(x) \) and its inverse \( f^{-1}(x) \), and then sketching both on a coordinate grid. It also includes determining if the inverse relation is a function. **Tables for Values:** - **\( x \) and \( f(x) \) Table:** - This table is for recording outputs \( f(x) \) for given inputs \( x \). - **\( x \) and \( f^{-1}(x) \) Table:** - This table is meant to record values of \( x \) with corresponding \( f^{-1}(x) \), the inverse relation. **Graph Details:** - The graph grid has axes ranging from -5 to 20 on the x-axis and -5 to 20 on the y-axis. - Students are expected to plot the quadratic function \( f(x) = -2x^2 + 18 \) and analyze its graph. - Additionally, plot the values for the inverse and explore the symmetry, particularly focusing on line \( y=x \) which can help visualize relation inverses. **Key Concepts:** - **Inverse Relation of a Function:** - For a quadratic like \( f(x) = -2x^2 + 18 \), the inverse may not be a function unless restricted on a specific interval because quadratics have a parabolic shape, which doesn’t pass the Horizontal Line Test. - **Determining if Inverse is a Function:** - Exploring if for each output there’s only one corresponding input in the inverse will be key. Graphically, the inverse would need to pass the Vertical Line Test if viewed as a function. --- This exercise assists in understanding function inverses, graphical representation, and conceptualizing whether certain mathematical inverses satisfy the criteria to be functions themselves.
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