Worksheet 7.4 Inverse Functions Answer Key Explained
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Inverse functions are a critical concept in mathematics that often confuse students, especially when it comes to understanding their properties and applications. Worksheet 7.4 focuses specifically on inverse functions, providing exercises that require students to determine whether certain functions have inverses and, if they do, to find the inverse functions. This article will explain the answer key for Worksheet 7.4, providing a comprehensive understanding of how to approach these problems.
What Are Inverse Functions? 🔄
Before diving into the specifics of the worksheet, it is essential to grasp what inverse functions are. An inverse function essentially "undoes" the action of a function. If we have a function ( f(x) ), its inverse ( f^{-1}(x) ) will take the output of ( f ) and return it to its original input.
Properties of Inverse Functions
To determine if a function has an inverse, it must be one-to-one (bijective), meaning it passes the horizontal line test. Here are some key properties:
- Injective (One-to-One): No two different inputs produce the same output.
- Surjective (Onto): Every element in the range is covered.
- The graphs of ( f(x) ) and ( f^{-1}(x) ) are reflections across the line ( y = x ).
Understanding Worksheet 7.4 📄
In Worksheet 7.4, students are presented with a set of functions and asked to determine if they are invertible. For those that are, students must find the inverse function. Here’s a breakdown of how to approach each type of problem.
Example Problems and Answers
Problem 1: Determining Inverses
Let’s say the first problem states ( f(x) = 2x + 3 ).
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Step 1: Check if ( f(x) ) is one-to-one:
- This function is linear with a slope of 2, so it passes the horizontal line test.
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Step 2: Find the inverse:
- To find the inverse, swap ( x ) and ( y ): [ y = 2x + 3 \implies x = 2y + 3 ]
- Solving for ( y ): [ y = \frac{x - 3}{2} ]
- Thus, ( f^{-1}(x) = \frac{x - 3}{2} ).
Problem 2: Not All Functions Have Inverses
Another problem could state ( g(x) = x^2 ) (for ( x \geq 0 )).
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Step 1: Check injectivity:
- The function ( g(x) ) fails the horizontal line test since two different inputs (e.g., 1 and -1) produce the same output (1).
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Conclusion:
- Since ( g(x) ) is not one-to-one, it does not have an inverse.
Table of Example Functions and Their Inverses
To clarify the findings from Worksheet 7.4, here’s a summary of some example functions and their inverses in a table format:
<table> <tr> <th>Function ( f(x) )</th> <th>Inverse ( f^{-1}(x) )</th> <th>Injective?</th> </tr> <tr> <td>2x + 3</td> <td>(x - 3)/2</td> <td>Yes</td> </tr> <tr> <td>x^2 (x ≥ 0)</td> <td>√x</td> <td>No (without restrictions)</td> </tr> <tr> <td>3x - 5</td> <td>(x + 5)/3</td> <td>Yes</td> </tr> <tr> <td>1/x</td> <td>1/y</td> <td>Yes (except at x=0)</td> </tr> </table>
Important Notes on Inverses 📝
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Restrictions: It’s important to note that some functions may only have inverses under specific conditions. For example, ( g(x) = x^2 ) is only invertible when restricted to non-negative numbers.
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Verification: To confirm that two functions are indeed inverses, check if ( f(f^{-1}(x)) = x ) and ( f^{-1}(f(x)) = x ).
Common Mistakes to Avoid 🚫
- Not Checking Injectivity: Always check if the function is one-to-one before attempting to find the inverse.
- Algebraic Errors: Be careful with algebraic manipulations when solving for the inverse.
- Ignoring Domain Restrictions: Remember to note any restrictions on the domain that might affect whether an inverse exists.
Conclusion
Worksheet 7.4 serves as a fundamental exercise in understanding inverse functions. By grasping the concepts of injectivity and the process for finding inverses, students can solidify their comprehension of this crucial mathematical concept. Remember to practice thoroughly and revisit the properties and definitions of inverse functions to enhance your understanding.