Inverse Functions
Henry Lee
7 min read
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Study Guide Overview
This study guide covers inverse functions, including: determining invertibility (one-to-one functions and horizontal line test), finding inverse functions (swap method), domains and ranges of inverse functions, graphing, and common exam question types. It also includes practice questions and an answer key.
#Inverse Functions: Your Night-Before-the-Test Guide 🚀
Hey there! Let's make sure you're totally ready to ace inverse functions. This guide is designed to be super clear and easy to use, especially when time is tight. Let’s dive in!
#What Makes a Function Invertible?
Before we get into the nitty-gritty, let’s nail down what makes a function invertible. It's all about these two key ideas:
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One-to-One Function: A function is one-to-one if each output (y-value) corresponds to only one input (x-value). Think of it like a perfect match – no repeats! 👯
- Horizontal Line Test: If any horizontal line crosses the graph more than once, it's not one-to-one. 📏
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Unrestricted Domain: The function's domain shouldn't be limited in a way that prevents it from having an inverse. It needs to cover all possible input values. 🌐
A function must be both one-to-one and have an unrestricted domain to be invertible. This is super important for both multiple choice and free response questions.
#Inverse Function Basics
- Notation: The inverse of a function f(x) is written as f⁻¹(x). ✍️
- Reflection: The graph of f⁻¹(x) is a reflection of f(x) across the line y = x. Imagine a mirror! 🪞
- Domain and Range Swap:
- The domain of f⁻¹(x) is the range of f(x).
- The range of f⁻¹(x) is the domain of f(x).
Think of an inverse function as the "undo" button for the original function. It reverses the input-output relationship. 🔄
#Input-Output Pairs
- If f(a) = b, then f⁻¹(b) = a. The input and output pairs are simply flipped! ...
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