Absolute Value And Step Functions Homework Answer Key

Absolute value and step functions are fundamental concepts in mathematics, especially when delving into algebra and calculus. Understanding these functions is crucial not only for academic success but also for grasping real-world applications in various fields. This comprehensive guide will provide a thorough exploration of absolute value and step functions, complete with examples, step-by-step solutions, and an answer key to help you master these topics.

Absolute Value Functions: A Comprehensive Overview

The absolute value of a number is its distance from zero on the number line, regardless of direction. This concept is simple yet powerful, forming the basis for more complex functions and applications.

Definition and Notation

The absolute value of a real number x is denoted as |x| and is defined as:

| x | = x, if x ≥ 0 | x | = -x, if x < 0

This means that if x is non-negative, the absolute value is x itself. If x is negative, the absolute value is the negation of x, which makes it positive.

Example:

• |5| = 5
• |-5| = -(-5) = 5
• |0| = 0

Properties of Absolute Value

Understanding the properties of absolute value is crucial for solving equations and inequalities involving absolute value functions.

1. Non-negativity: |x| ≥ 0 for all real numbers x.
2. Symmetry: |x| = |- x| for all real numbers x.
3. Product: |xy| = |x| |y| for all real numbers x and y.
4. Quotient: |x/y| = |x| / |y| for all real numbers x and y, where y ≠ 0.
5. Triangle Inequality: |x + y| ≤ |x| + |y| for all real numbers x and y.

Graphing Absolute Value Functions

The simplest absolute value function is f(x) = |x|. Its graph is V-shaped, with the vertex at the origin (0, 0). For x ≥ 0, the graph is the same as y = x, and for x < 0, the graph is y = -x.

Transformations of Absolute Value Functions:

• Vertical Shift: f(x) = |x| + k shifts the graph vertically.
• Horizontal Shift: f(x) = |x - h| shifts the graph horizontally.
• Vertical Stretch/Compression: f(x) = a|x| stretches or compresses the graph vertically.
• Reflection: f(x) = -|x| reflects the graph across the x-axis.

Example:

Graph f(x) = |x - 2| + 3.

• This is the graph of f(x) = |x| shifted 2 units to the right and 3 units up. The vertex is at (2, 3).

Solving Absolute Value Equations

To solve an absolute value equation |f(x)| = a, where a ≥ 0, we set up two equations:

1. f(x) = a
2. f(x) = -a

Example 1:

Solve |2x - 1| = 5.

1. 2x - 1 = 5 => 2x = 6 => x = 3
2. 2x - 1 = -5 => 2x = -4 => x = -2

Thus, the solutions are x = 3 and x = -2.

Example 2:

Solve |3x + 2| = -1.

• Since the absolute value cannot be negative, there is no solution.

Solving Absolute Value Inequalities

Solving absolute value inequalities involves considering two cases, similar to equations.

1. |f(x)| < a (where a > 0)

   This is equivalent to -a < f(x) < a.

2. |f(x)| > a (where a > 0)

   This is equivalent to f(x) < -a or f(x) > a.

Example 1:

Solve |x - 3| < 2.

• -2 < x - 3 < 2
• Add 3 to all parts: 1 < x < 5

The solution is the interval (1, 5).

Example 2:

Solve |2x + 1| > 3.

1. 2x + 1 < -3 => 2x < -4 => x < -2
2. 2x + 1 > 3 => 2x > 2 => x > 1

The solution is x < -2 or x > 1. In interval notation, this is (-∞, -2) ∪ (1, ∞).

Step Functions: An In-Depth Exploration

Step functions, also known as staircase functions, are piecewise constant functions that jump to different constant values over specific intervals. They are frequently encountered in discrete mathematics and signal processing.

Definition and Types of Step Functions

A step function is a piecewise function defined by constant values over intervals. The most common types of step functions are:

1. Greatest Integer Function (Floor Function): Denoted as f(x) = ⌊x⌋, it returns the largest integer less than or equal to x.

   Example: ⌊3.7⌋ = 3, ⌊-2.3⌋ = -3

2. Least Integer Function (Ceiling Function): Denoted as f(x) = ⌈x⌉, it returns the smallest integer greater than or equal to x.

   Example: ⌈3.7⌉ = 4, ⌈-2.3⌉ = -2

3. Heaviside Step Function (Unit Step Function): Denoted as H(x), it is defined as:

   H(x) = 0, if x < 0 H(x) = 1, if x ≥ 0

Properties of Step Functions

1. Discontinuity: Step functions are discontinuous at integer values for floor and ceiling functions, and at x = 0 for the Heaviside function.
2. Piecewise Constant: Within each interval between integers, the function is constant.
3. Non-Continuous Derivatives: The derivative is zero wherever the function is continuous but undefined at points of discontinuity.

Graphing Step Functions

The graph of a step function looks like a series of horizontal line segments, with jumps at certain points.

Graphing the Greatest Integer Function (Floor Function):

• For any interval [n, n+1), where n is an integer, the value of the function is n.
• The graph consists of horizontal line segments at integer heights.

Graphing the Least Integer Function (Ceiling Function):

• For any interval (n, n+1], where n is an integer, the value of the function is n+1.
• The graph consists of horizontal line segments at integer heights, one unit higher than the floor function.

Graphing the Heaviside Step Function:

• The function is 0 for x < 0 and jumps to 1 at x = 0.
• It remains 1 for all x ≥ 0.

Applications of Step Functions

1. Computer Science: Representing digital signals, rounding functions.
2. Signal Processing: Modeling on/off signals, sampling.
3. Economics: Modeling tax brackets, pricing strategies.
4. Physics: Describing ideal switches, instantaneous changes.

Examples of Step Functions

Example 1:

Graph f(x) = ⌊x/2⌋.

• This function scales the input x by a factor of 1/2 and then takes the floor.
• The steps occur at even integers.

Example 2:

Graph f(x) = 2⌈x⌉ - 1.

• This function takes the ceiling of x, multiplies it by 2, and subtracts 1.
• The steps occur at integer values, and the height of each step is 2.

Example 3:

Graph f(x) = H(x - 3).

• This function is a Heaviside step function shifted 3 units to the right.
• It is 0 for x < 3 and 1 for x ≥ 3.

Absolute Value and Step Functions: Homework Answer Key

This section provides an answer key for homework problems involving absolute value and step functions. The solutions are detailed to help you understand the step-by-step process.

Absolute Value Equations and Inequalities

Problem 1:

Solve |4x - 3| = 7.

Solution:

1. 4x - 3 = 7 => 4x = 10 => x = 10/4 = 5/2
2. 4x - 3 = -7 => 4x = -4 => x = -1

Answer: x = 5/2, x = -1

Problem 2:

Solve |x + 5| ≤ 3.

Solution:

• -3 ≤ x + 5 ≤ 3
• Subtract 5 from all parts: -8 ≤ x ≤ -2

Answer: [-8, -2]

Problem 3:

Solve |3 - 2x| > 5.

Solution:

1. 3 - 2x < -5 => -2x < -8 => x > 4
2. 3 - 2x > 5 => -2x > 2 => x < -1

Answer: x < -1 or x > 4

Problem 4:

Solve |x - 4| + 2 = 6.

Solution:

• |x - 4| = 4

1. x - 4 = 4 => x = 8
2. x - 4 = -4 => x = 0

Answer: x = 8, x = 0

Problem 5:

Solve |5x + 2| = 0.

Solution:

• 5x + 2 = 0 => 5x = -2 => x = -2/5

Answer: x = -2/5

Step Functions

Problem 1:

Evaluate ⌊-3.6⌋ + ⌈4.2⌉.

Solution:

• ⌊-3.6⌋ = -4
• ⌈4.2⌉ = 5
• -4 + 5 = 1

Answer: 1

Problem 2:

Evaluate H(-2) + H(0) + H(5).

Solution:

• H(-2) = 0
• H(0) = 1
• H(5) = 1
• 0 + 1 + 1 = 2

Answer: 2

Problem 3:

Graph f(x) = ⌊x⌋ + 1 for -3 ≤ x ≤ 3.

Solution:

• The function is a floor function shifted up by 1 unit.
• The graph consists of horizontal line segments at integer heights, with open circles at the right endpoints.

Problem 4:

Graph f(x) = 2H(x + 1) - 1.

Solution:

• The function is a Heaviside step function shifted 1 unit to the left, scaled by a factor of 2, and shifted down by 1 unit.
• For x < -1, f(x) = -1
• For x ≥ -1, f(x) = 1

Problem 5:

Determine the value of ⌈x⌉ - ⌊x⌋ when x is an integer.

Solution:

• If x is an integer, ⌈x⌉ = x and ⌊x⌋ = x.
• Therefore, ⌈x⌉ - ⌊x⌋ = x - x = 0

Answer: 0

Advanced Applications and Problem Solving

To further solidify your understanding, let's delve into more advanced applications and problem-solving techniques.

Advanced Absolute Value Problems

Problem 1:

Find all values of x such that |x - 1| = |2x + 3|.

Solution:

We have two cases:

1. x - 1 = 2x + 3 => -x = 4 => x = -4
2. x - 1 = -(2x + 3) => x - 1 = -2x - 3 => 3x = -2 => x = -2/3

Answer: x = -4, x = -2/3

Problem 2:

Solve the inequality |x - 2| < |x + 1|.

Solution:

We can solve this by squaring both sides to eliminate the absolute value signs:

(x - 2)^2 < (x + 1)^2 x^2 - 4x + 4 < x^2 + 2x + 1 -6x < -3 x > 1/2

Answer: x > 1/2

Problem 3:

Sketch the graph of y = ||x| - 1|.

Solution:

1. First, graph y = |x|.
2. Then, shift it down by 1 unit to get y = |x| - 1.
3. Finally, take the absolute value of the entire expression, reflecting any parts of the graph below the x-axis back above.

The resulting graph is W-shaped.

Advanced Step Function Problems

Problem 1:

Evaluate ∫05 ⌊x⌋ dx.

Solution:

We can break the integral into intervals where the floor function is constant:

∫05 ⌊x⌋ dx = ∫01 0 dx + ∫12 1 dx + ∫23 2 dx + ∫34 3 dx + ∫45 4 dx = 0 + (1)(1) + (2)(1) + (3)(1) + (4)(1) = 0 + 1 + 2 + 3 + 4 = 10

Answer: 10

Problem 2:

Express the function f(x) = { 0, if x < 2; 3, if x ≥ 2 } using the Heaviside step function.

Solution:

f(x) = 3H(x - 2)

Problem 3:

Graph f(x) = ⌊x⌋ - ⌈x⌉.

Solution:

• If x is an integer, then ⌊x⌋ = ⌈x⌉ = x, so f(x) = 0.
• If x is not an integer, then ⌈x⌉ = ⌊x⌋ + 1, so f(x) = -1.

The graph is 0 at integer values and -1 everywhere else.

Conclusion

Mastering absolute value and step functions is crucial for a solid foundation in mathematics. These functions appear in various applications, from basic algebra to advanced calculus and beyond. By understanding their definitions, properties, and applications, you can confidently tackle a wide range of problems. This comprehensive guide, complete with examples, step-by-step solutions, and an answer key, is designed to help you achieve mastery and excel in your studies. Remember to practice regularly and apply these concepts to real-world scenarios to reinforce your understanding.