Master

Functions: Graphs and Intersections

Master functions: graphs and intersections with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Functions: Graphs and Intersections

Study Guide

Key Definition

The intersection points of two functions $f(x)$ and $g(x)$ are the $x$ values where $f(x) = g(x)$.

Important Notes

To find intersections algebraically, solve $f(x) = g(x)$.
Graphical solutions involve plotting and identifying where the graphs intersect.
For complex functions, technology may be needed to estimate intersection points.
Absolute value and logarithmic functions can create non-trivial intersections.
Understanding graph behavior is crucial for visualizing intersections.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

$\div$ represents division

$|x|$ denotes absolute value

$\log(x)$ denotes logarithm

Remember to use proper notation when solving problems

Why It Works

Intersection points are solutions to $f(x) = g(x)$, where their graphs meet.

Remember

Intersection points can be found algebraically or graphically using $f(x) = g(x)$.

Quick Reference

Intersection Equation:$f(x) = g(x)$

Understanding Functions: Graphs and Intersections

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

For $f(x) = x + 6$ and $g(x) = -x$, set $x + 6 = -x$. Solving gives $x = -3$ and $y = f(-3) = 3$, so they intersect at $(-3, 3)$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the solution to $x + 6 = -x$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine two functions representing two friends' distances from a park as $f(x) = x + 6$ and $g(x) = -x$. Find when they meet.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine the intersection of $f(x) = |x - 5|$ and $g(x) = \log(x)$.

Challenge Quiz

Single Choice Quiz

Advanced

Solve for the intersection of $f(x) = x^2$ and $g(x) = 4 - x^2$.

Recap

Watch & Learn

Review key concepts and takeaways