IB Mathematics AA – Topic 2: Functions

Comprehensive Guide to Modulus Functions & Inequalities

Introduction to Modulus Functions

The modulus function (also called the absolute value function) measures the distance of a number from zero on the number line, regardless of direction. It always produces a non-negative output, making it essential for modeling situations involving magnitude, distance, error, and tolerance.

Key concept: The modulus of \(x\), written as \(|x|\), gives the "size" or "magnitude" of \(x\) without considering its sign. Whether \(x\) is positive or negative, \(|x|\) is always non-negative. This seemingly simple function has profound applications in optimization, physics, engineering, and computer science.

Why modulus matters: Modulus functions create V-shaped graphs with distinct properties. Understanding how to graph, transform, and solve equations and inequalities involving modulus is crucial for analyzing piecewise functions, absolute deviations, and error bounds in real-world applications.

In this guide: We'll master the definition and properties of modulus functions, learn to graph transformations of modulus functions, develop systematic techniques for solving modulus equations (often requiring case-by-case analysis), and tackle modulus inequalities using both algebraic and graphical methods essential for IB exams.

1. The Modulus Function

Definition

Modulus Function Definition

\(|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}\)

The modulus gives the non-negative value of \(x\)

Key Properties

Essential Properties:

1. Non-negativity: \(|x| \geq 0\) for all real \(x\)
2. Symmetry: \(|-x| = |x|\) (even function)
3. Triangle Inequality: \(|x + y| \leq |x| + |y|\)
4. Product: \(|xy| = |x| \cdot |y|\)
5. Quotient: \(\left|\frac{x}{y}\right| = \frac{|x|}{|y|}\) for \(y \neq 0\)

Graph of Modulus Function

For \(f(x) = |x|\):

Domain: All real numbers, \(x \in \mathbb{R}\)
Range: \(y \geq 0\) or \([0, \infty)\)
Shape: V-shaped graph with vertex at origin \((0, 0)\)
Symmetry: Symmetric about y-axis (even function)
Gradient: -1 for \(x < 0\); +1 for \(x > 0\); undefined at \(x = 0\)
Intercepts: x-intercept and y-intercept both at \((0, 0)\)

Transformations of Modulus

Two Important Cases:

Type 1: \(y = |f(x)|\)

Take the graph of \(f(x)\) and reflect any part below the x-axis upward

All negative y-values become positive

Example: \(y = |x - 2|\) reflects negative portions up

Type 2: \(y = f(|x|)\)

Keep the right side of graph (\(x \geq 0\)) and reflect it across the y-axis

Graph becomes symmetric about y-axis

Example: \(y = (|x|)^2 = x^2\) (already even, so unchanged)

⚠ Common Pitfalls:

Forgetting negative case: \(|x|\) has two cases—don't forget \(x < 0\)!
Sign confusion: When \(x < 0\), \(|x| = -x\) (which is positive!)
Graph transformation: \(|f(x)|\) vs \(f(|x|)\) produce different graphs
Range restriction: \(|f(x)|\) always has range \(\geq 0\)

2. Solving Modulus Equations

Basic Equation Types

Three Standard Forms:

Form 1: \(|x| = a\)

If \(a > 0\): Two solutions, \(x = a\) or \(x = -a\)
If \(a = 0\): One solution, \(x = 0\)
If \(a < 0\): No solution (modulus cannot be negative)

Example: \(|x| = 5\) → \(x = 5\) or \(x = -5\)

Form 2: \(|f(x)| = a\)

Solve two equations: \(f(x) = a\) and \(f(x) = -a\)

Example: \(|x - 3| = 7\) → \(x - 3 = 7\) or \(x - 3 = -7\) → \(x = 10\) or \(x = -4\)

Form 3: \(|f(x)| = |g(x)|\)

Solve: \(f(x) = g(x)\) or \(f(x) = -g(x)\)

Example: \(|x + 1| = |2x - 3|\)

Case-by-Case Method

For Complex Modulus Equations:

Identify critical points where expression inside modulus equals zero
Split the domain into regions based on these points
In each region, remove the modulus sign (positive or negative)
Solve the resulting equation in each region
Check solutions are valid for their region
Combine all valid solutions

💡 Solving Tips:

Always check your solutions by substituting back
Remember \(|x| = a\) requires \(a \geq 0\)
Square both sides only if both sides are already positive (to avoid extraneous solutions)
Graphical method: find intersections of \(y = |f(x)|\) and \(y = a\)

Example 1: Solving Modulus Equations

Problem: Solve the following equations:

(a) \(|2x - 6| = 10\)

(b) \(|x - 1| = |2x + 3|\)

Solution:

(a) \(|2x - 6| = 10\)

This has form \(|f(x)| = a\) where \(a = 10 > 0\)

Solve two equations:

Case 1: \(2x - 6 = 10\)

\(2x = 16\)

\(x = 8\)

Case 2: \(2x - 6 = -10\)

\(2x = -4\)

\(x = -2\)

Check both solutions:

\(x = 8\): \(|2(8) - 6| = |10| = 10\) ✓

\(x = -2\): \(|2(-2) - 6| = |-10| = 10\) ✓

Solutions: \(x = 8\) or \(x = -2\)

(b) \(|x - 1| = |2x + 3|\)

This has form \(|f(x)| = |g(x)|\)

Solve two equations:

Case 1: \(x - 1 = 2x + 3\)

\(-1 - 3 = 2x - x\)

\(x = -4\)

Case 2: \(x - 1 = -(2x + 3)\)

\(x - 1 = -2x - 3\)

\(3x = -2\)

\(x = -\frac{2}{3}\)

Check both solutions:

\(x = -4\): \(|-5| = |2(-4) + 3| = |-5|\) → \(5 = 5\) ✓

\(x = -\frac{2}{3}\): \(|-\frac{5}{3}| = |\frac{-4}{3} + 3| = |\frac{5}{3}|\) → \(\frac{5}{3} = \frac{5}{3}\) ✓

Solutions: \(x = -4\) or \(x = -\frac{2}{3}\)

3. Solving Modulus Inequalities

Standard Inequality Forms

Key Inequality Rules

Type 1: \(|f(x)| < a\) (where \(a > 0\))

\(-a < f(x) < a\)

"The solution is between \(-a\) and \(a\)"

Example: \(|x| < 3\) → \(-3 < x < 3\)

Type 2: \(|f(x)| > a\) (where \(a > 0\))

\(f(x) < -a\) or \(f(x) > a\)

"The solution is outside the interval"

Example: \(|x| > 3\) → \(x < -3\) or \(x > 3\)

Special Cases:

If \(a \leq 0\) and \(|f(x)| < a\): No solution
If \(a < 0\) and \(|f(x)| > a\): All real numbers (since \(|f(x)| \geq 0\))

Solution Methods

Three Approaches:

Method 1: Algebraic (using inequalities above)

Apply the standard forms directly

Method 2: Case-by-case

Split into regions based on where expression = 0

Method 3: Graphical

Sketch \(y = |f(x)|\) and \(y = a\), find where inequality satisfied

⚠ Critical Mistakes:

Wrong direction: \(|x| > 3\) is NOT \(-3 > x > 3\) (that's impossible!)
Forgetting "or": \(|x| > 3\) means \(x < -3\) OR \(x > 3\) (two separate regions)
Sign flipping: When multiplying/dividing by negative, flip inequality sign
Not checking validity: Solutions must satisfy original inequality

Example 2: Solving Modulus Inequalities (IB-Style)

Problem: Solve the following inequalities:

(a) \(|x - 5| \leq 3\)

(b) \(|2x + 1| > 7\)

Solution:

(a) \(|x - 5| \leq 3\)

This has form \(|f(x)| \leq a\) where \(a = 3 > 0\)

Apply the rule: \(-a \leq f(x) \leq a\)

\(-3 \leq x - 5 \leq 3\)

Add 5 to all parts:

\(-3 + 5 \leq x \leq 3 + 5\)

\(2 \leq x \leq 8\)

Solution: \(x \in [2, 8]\) or \(2 \leq x \leq 8\)

Interpretation: \(x\) is within distance 3 from 5 on the number line

(b) \(|2x + 1| > 7\)

This has form \(|f(x)| > a\) where \(a = 7 > 0\)

Apply the rule: \(f(x) < -a\) or \(f(x) > a\)

\(2x + 1 < -7\) or \(2x + 1 > 7\)

Case 1: \(2x + 1 < -7\)

\(2x < -8\)

\(x < -4\)

Case 2: \(2x + 1 > 7\)

\(2x > 6\)

\(x > 3\)

Solution: \(x < -4\) or \(x > 3\)

Interval notation: \(x \in (-\infty, -4) \cup (3, \infty)\)

Interpretation: \(x\) is more than distance 7 from \(-\frac{1}{2}\) on the number line

📋 Modulus Quick Reference

Type	Statement	Solution
Equation	\(\|f(x)\| = a\), \(a > 0\)	\(f(x) = a\) or \(f(x) = -a\)
Inequality <	\(\|f(x)\| < a\), \(a > 0\)	\(-a < f(x) < a\)
Inequality >	\(\|f(x)\| > a\), \(a > 0\)	\(f(x) < -a\) or \(f(x) > a\)
Double modulus	\(\|f(x)\| = \|g(x)\|\)	\(f(x) = g(x)\) or \(f(x) = -g(x)\)

🎯 IB Exam Strategy

Common Question Types:

"Solve \(|f(x)| = a\)": Two cases—positive and negative
"Solve \(|f(x)| < a\)": Double inequality \(-a < f(x) < a\)
"Solve \(|f(x)| > a\)": Two separate regions (OR statement)
"Sketch \(y = |f(x)|\)": Reflect negative portions upward
"Find where \(|f(x)| \leq a\)": Can use graphical interpretation

Key Reminders:

Always check: does \(a \geq 0\)? (modulus cannot equal negative)
For inequalities: "less than" gives AND, "greater than" gives OR
Use GDC to verify graphically
Check all solutions in original equation/inequality

🎉 Master Modulus Functions!

Modulus functions represent distance and magnitude—fundamental concepts in mathematics. Mastering their definition, graphical transformations, and systematic solution methods for equations and inequalities gives you powerful analytical tools essential for IB success and real-world applications!

Key Success Factors:

✓ Definition: \(|x| = x\) if \(x \geq 0\); \(|x| = -x\) if \(x < 0\)
✓ Equation \(|f(x)| = a\): solve \(f(x) = a\) and \(f(x) = -a\)
✓ Inequality \(|f(x)| < a\): gives \(-a < f(x) < a\) (AND)
✓ Inequality \(|f(x)| > a\): gives \(f(x) < -a\) or \(f(x) > a\) (OR)
✓ Graph \(y = |f(x)|\): reflect negative parts upward
✓ Always verify solutions work in original problem

Understand Distance • Split Into Cases • Check Solutions

Master modulus and excel in IB Mathematics! 🚀