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Even/Odd Functions

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Even/Odd Functions

Study Guide

Key Definition

A function is even if $f(-x) = f(x)$ for all x in its domain. A function is odd if $f(-x) = -f(x)$ for all x in its domain.

Important Notes

Even functions have symmetry about the $y$-axis.
Odd functions have rotational symmetry about the origin.
Not all functions are even or odd.
Check the symmetry to determine the type.
Test with $f(-x)$ to see if a function is even or odd.

Mathematical Notation

$f(-x)$ means the function value at negative $x$

$f(x)$ is the original function value

$+$ represents addition

$-$ represents subtraction

Remember to use proper notation when solving problems

Why It Works

Understanding symmetry helps determine if a function is even or odd. Even functions reflect across the $y$-axis and odd functions rotate around the origin.

Remember

Check $f(-x)$ to see if it equals $f(x)$ or $-f(x)$.

Quick Reference

Even Function:$f(-x) = f(x)$

Odd Function:$f(-x) = -f(x)$

Understanding Even/Odd Functions

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Even functions satisfy $f(-x) = f(x)$. Odd functions satisfy $f(-x) = -f(x)$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Determine if $f(x) = x^2$ is even, odd, or neither.

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboard ramp is symmetric about the center. Test if its equation $f(x) = x^3$ is even or odd.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Is $f(x) = x^2 + 1$ even, odd, or neither?

Challenge Quiz

Single Choice Quiz

Advanced

Determine if $f(x) = x^3 + x$ is even, odd, or neither.

Recap

Watch & Learn

Review key concepts and takeaways