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Inverse Functions

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Inverse Functions

Study Guide

Key Definition

An inverse function, denoted as $f^{-1}(x)$, reverses the effect of the original function $f(x)$.

Important Notes

The inverse function is only defined for one-to-one functions.
In notation, $f^{-1}(x)$ does not mean 1/f(x).
To find an inverse, swap the roles of $x$ and $y$ in the equation $y = f(x)$.
Verify inverses by checking if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Real-world applications include physics and engineering.

Mathematical Notation

$f^{-1}(x)$ means inverse function of $f(x)$

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Inverse functions work because they undo the effect of the original function, allowing us to retrieve the input from the output, effectively reversing the process.

Remember

To find the inverse, exchange $x$ and $y$ in the function equation and solve for $y$.

Quick Reference

One-to-One Function:Each $x$ value maps to a unique $y$ value.

Inverse Notation:$f^{-1}(x)$

Understanding Inverse Functions

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

An inverse function reverses the effect of the original function, such that $f(f^{-1}(x)) = x$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the inverse of the function $f(x) = 2x + 3$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You are using a function to calculate the price of apples based on weight: $f(x) = 3x + 2$. Find the inverse to determine the weight of apples if you know the price.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Challenge yourself to find the inverse of the quadratic function $f(x) = x^2$ for $x \geq 0$.

Challenge Quiz

Single Choice Quiz

Advanced

What is the inverse of the function $f(x) = \frac{2x + 3}{x - 1}$?

Recap

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