Grade 11 Mathematics Unit 2 : Rational Expressions and Rational Functions

Unit 2, “Rational Expressions and Rational Functions,” builds upon your knowledge of polynomials by exploring expressions and functions that are structured as fractions. This unit covers the essential skills for manipulating, solving, and graphing these important mathematical objects.

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Chapter 2.1 – Rational Expressions

This chapter covers the algebra of polynomial fractions.

• Define a rational expression as a fraction where the numerator and denominator are both polynomials, with the key rule that the denominator cannot equal zero.
• Learn to find the domain of a rational expression by identifying and excluding any values of the variable that would make the denominator zero.
• Master simplifying rational expressions by factoring both the numerator and denominator and canceling any common factors.
• Practice operations on rational expressions, including multiplication, division, addition, and subtraction.

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Chapter 2.2 – Rational Equations and Rational Inequalities

Here, you’ll apply your skills to solve complex problems involving rational expressions.

• Learn to solve rational equations by multiplying every term by the Least Common Denominator (LCD) to eliminate the fractions.
• Remember to always check for extraneous solutions—answers that don’t work in the original equation because they make a denominator zero.
• Develop a step-by-step method for solving rational inequalities using a sign chart to test intervals and find the correct solution set.

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Chapter 2.3 – Rational Functions and Their Graphs

This section explores the unique visual characteristics of rational functions.

• Analyze the key features of rational function graphs, especially asymptotes, which are lines the graph gets closer and closer to but never touches.
• Learn to identify vertical asymptotes (where the denominator is zero) and horizontal asymptotes (determined by comparing the degrees of the polynomials).
• Find x- and y-intercepts and identify any holes in the graph to create an accurate sketch.

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Chapter 2.4 – Applications

See how rational equations are used to model and solve real-life problems.

• Solve classic work-rate problems (e.g., two people working together to complete a job).
• Tackle problems involving distance, rate, and time, especially with variables like wind or water currents.

The unit finishes with a Summary and Review Exercise to sharpen your new skills.

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Learning Outcomes

By the end of this unit, you will be able to:

• Define and recognize rational expressions and functions.
• Simplify complex rational expressions using various methods.
• Solve rational equations and inequalities efficiently.
• Understand the key concepts of rational functions, including their graphs and asymptotes.