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Rational Functions

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Rational Functions

Study Guide

Key Definition

A rational function is a function that can be expressed as the ratio of two polynomials $\frac{p(x)}{q(x)}$ where $q(x) \neq 0$.

Important Notes

A rational function is undefined when $q(x) = 0$.
The numerator and denominator of a rational function are polynomials.
Simplification may be necessary to reveal the true nature of a rational function.
Understanding the behavior near the zeros of $q(x)$ is crucial.
Graphical analysis can provide insights into the function's behavior.

Mathematical Notation

$\frac{a}{b}$ means a divided by b

$x^2$ represents x squared

$\sqrt{x}$ represents the square root of x

$f(x)$ represents a function of x

Remember to use proper notation when solving problems

Why It Works

The function $\frac{p(x)}{q(x)}$ is defined as long as $q(x) \neq 0$, ensuring valid division and meaningful results.

Remember

Always check for values that make $q(x) = 0$ to identify undefined points.

Quick Reference

Rational Function:$\frac{p(x)}{q(x)}$

Undefined Condition:$q(x) = 0$

Understanding Rational Functions

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A rational function looks like $\frac{x^2 + 5x + 2}{4x + 7}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following is a rational function? $\frac{x^2 + 1}{x - 1}$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

If a company produces $p(x) = x^2 + 3x + 2$ products at cost $q(x) = 2x + 5$, find and simplify the average cost per product.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Why is the function $\frac{x^2 + 1}{x - 3}$ undefined at $x = 3$?

Challenge Quiz

Single Choice Quiz

Advanced

Evaluate the limit of $\frac{x^2 - 1}{x - 1}$ as $x$ approaches $1$.

Recap

Watch & Learn

Review key concepts and takeaways