Lesson 2.4: The Modulus Function and Inequalities

Introduction

Welcome to Lesson 2.4 where we'll dive into the fascinating world of the modulus function! The objectives of this lesson are designed to help you understand the concept of modulus and how it operates within the realm of inequalities. By the end of this lesson, you should be able to:

• Define and graph $|x|$ and $|f(x)|$.
• Solve equations and inequalities that involve the modulus.
• Sketch transformations that include moduli.
• Compare algebraic and graphical methods for solving modulus inequalities.
• Illustrate $y = |f(x)|$ from the function $y = f(x)$.

Ready? Let’s go! 🎉

Understanding the Modulus Function

The modulus function, denoted as $|x|$, is a way to represent the absolute value of $x$. It outputs the distance of $x$ from 0 on the number line, regardless of its sign. In simple terms:

• If $x \geq 0$, then $|x| = x$.
• If $x < 0$, then $|x| = -x$.

For example:

• $|5| = 5$
• $|-3| = 3$

This means that the graph of $y = |x|$ has a V shape, as it consists of two linear pieces:

• The line $y = x$ for $x \geq 0$.
• The line $y = -x$ for $x < 0$.

Graphing $|x|$

To graph $y = |x|$:

1. Plot points when $x$ is positive: (1,1), (2,2), (3,3).
2. Plot points when $x$ is negative: (-1,1), (-2,2), (-3,3).

Connecting these points creates the characteristic V shape. Let's take it up a notch by considering $|f(x)|$ where $f(x)$ is any function, say $f(x) = x^2 - 4$. The function $f(x)$ has roots, where it crosses the x-axis, and the modulus graph will reflect any parts below the x-axis upwards.

Solving Modulus Equations

Now, let’s tackle equations involving the modulus. For instance, if we have the equation:

$$|x - 3| = 5,$$

this means we are looking for $ x $ such that the distance from $ 3 $ is $ 5 $. This translates into two possible cases:

1. $x - 3 = 5$
2. $x - 3 = -5$

Solving these gives:

• Case 1: $x = 8$
• Case 2: $x = -2$

Thus, the solutions are $ x = 8 $ and $ x = -2 $. 🔍

Modulus Inequalities

Modulus inequalities can also be interesting! Let's solve:

$$|x - 4| < 3.$$

This inequality states that the distance from $ 4 $ is less than $ 3 $. For modulus inequalities, we often break it down into two cases:

1. $x - 4 < 3$
2. $x - 4 > -3$

The solutions become:

• From case 1: $ x < 7 $
• From case 2: $ x > 1 $

So, the solution to the inequality is:

$$1 < x < 7.$$

Sketching Transformations Involving Modulus

Transformations can include shifts, stretches, or reflections. For example, consider the function:

$$y = |x - 2| + 3.$$

This represents a shift of the graph of $y = |x|$:

• Shifted to the right by $ 2 $ units.
• Shifted up by $ 3 $ units.

The V shape will now appear with its vertex at $ (2, 3) $. You can sketch it by first graphing $y = |x|$ and then applying the transformations. 📈

Algebraic vs Graphical Methods for Modulus Inequalities

Algebraically, we use the methods discussed above, whereas graphically, we can visualize how the modulus affects a function. For example, to solve:

$$|x + 1| > 2,$$

we can again consider two cases:

1. $x + 1 > 2$
2. $x + 1 < -2$

This yields:

• From case 1: $ x > 1 $
• From case 2: $ x < -3 $

Thus, the solution for this inequality is:

$$x < -3 \quad \text{or} \quad x > 1.$$

In comparison, the graphical approach would reveal two regions on a graph where the function lies above the x-axis, corresponding to the same solution. 🖌️

Sketching $y = |f(x)|$ from $y = f(x)$

To illustrate the effect of modulus on some continuous function $ f(x) $, let’s say:

$$f(x) = x^2 - 4$$

This function equals zero at points $ x = -2 $ and $ x = 2 $. Below the x-axis, $y = f(x)$ becomes negative, and thus $y = | f(x) |$ will be reflected upwards:

• Between $ -2 $ and $ 2 $, where $ f(x) $ is negative, $y = |f(x)|$ becomes positive.
• For $ x > 2 $ and $ x < -2$, the graph remains the same as $ f(x) $.

Conclusion

In this lesson, we explored key concepts around the modulus function and inequalities. We learned how to define, graph, and solve equations and inequalities involving modulus. Additionally, we examined how to sketch transformations and compare methods for solving modulus inequalities. You should now feel more confident with modulus functions and their applications! 💡

Study Notes

• The modulus function $|x|$ represents the absolute value of $x$.
• Solving $|x - a| = b$ leads to two cases: $x - a = b$ and $x - a = -b$.
• For modulus inequalities $|x - a| < b$, solve by creating two inequalities: $x - a < b$ and $x - a > -b$.
• Transformations include shifts, reflections, and stretches involving the modulus.
• Graphing $y = |f(x)|$ involves reflecting sections of $y = f(x)$ that are below the x-axis upwards.