Absolute Value And Step Functions Homework Answer Key

Absolute value and step functions form a crucial part of mathematics, especially in algebra and calculus. Mastering these concepts often requires solving numerous problems, and having a reliable answer key can significantly aid in understanding the underlying principles and verifying your solutions. This article provides an in-depth exploration of absolute value and step functions, covering their definitions, properties, applications, and common problem-solving techniques, accompanied by a comprehensive answer key to guide you through your homework.

Understanding Absolute Value Functions

The absolute value of a number x, denoted as |x|, represents the distance of x from zero on the number line. Essentially, it converts any negative number into its positive counterpart, while leaving positive numbers and zero unchanged.

Definition and Properties

The formal definition of the absolute value function is:

|x| = x, if x ≥ 0 -x, if x < 0

Key properties of absolute value functions include:

• Non-negativity: |x| ≥ 0 for all x.
• Symmetry: |x| = |-x| for all x.
• Product Rule: |xy| = |x||y| for all x and y.
• Quotient Rule: |x/y| = |x|/|y| for all x and y ≠ 0.
• Triangle Inequality: |x + y| ≤ |x| + |y| for all x and y.

Graphing Absolute Value Functions

The simplest absolute value function, f(x) = |x|, has a V-shaped graph with the vertex at the origin (0,0). The graph is symmetric about the y-axis due to the property |x| = |-x|.

Transformations of absolute value functions can be represented as f(x) = a|x - h| + k, where:

• a stretches or compresses the graph vertically. If a < 0, the graph is reflected over the x-axis.
• h shifts the graph horizontally.
• k shifts the graph vertically.

Solving Absolute Value Equations and Inequalities

Solving absolute value equations and inequalities involves considering two cases: the expression inside the absolute value is either positive or negative.

• Equations: To solve |f(x)| = c, where c is a constant, solve both f(x) = c and f(x) = -c.
• Inequalities:
  • To solve |f(x)| < c, solve -c < f(x) < c.
  • To solve |f(x)| > c, solve f(x) < -c or f(x) > c.

Exploring Step Functions

Step functions, also known as staircase functions, are piecewise constant functions that exhibit abrupt changes in value. The most common example is the greatest integer function, also known as the floor function.

Definition and Types

A step function is a function that is piecewise constant, meaning it consists of horizontal line segments. Key types of step functions include:

• Greatest Integer Function (Floor Function): Denoted as ⌊x⌋, it returns the largest integer less than or equal to x. For example, ⌊3.7⌋ = 3, ⌊-2.3⌋ = -3, and ⌊5⌋ = 5.
• Least Integer Function (Ceiling Function): Denoted as ⌈x⌉, it returns the smallest integer greater than or equal to x. For example, ⌈3.7⌉ = 4, ⌈-2.3⌉ = -2, and ⌈5⌉ = 5.
• Heaviside Step Function: Denoted as H(x), it is defined as:

H(x) = 0, if x < 0 1, if x ≥ 0

Properties of Step Functions

• Discontinuity: Step functions are discontinuous at integer values (for floor and ceiling functions) or at a specific point (for the Heaviside function).
• Piecewise Constant: The function's value remains constant between integer values.
• Non-decreasing: The floor and ceiling functions are non-decreasing, meaning their values never decrease as x increases.

Graphing Step Functions

The graph of a step function consists of horizontal line segments (steps) with jumps at specific points. For the floor function, the steps are closed on the left and open on the right, indicating that the function takes the integer value at the left endpoint of each interval.

Applications of Step Functions

Step functions have numerous applications in mathematics, computer science, and engineering:

• Computer Science: Representing digital signals, rounding numbers, and implementing conditional statements.
• Engineering: Modeling on/off switches, signal processing, and control systems.
• Mathematics: Approximating integrals, defining piecewise functions, and analyzing discrete systems.

Absolute Value and Step Functions: Homework Answer Key

This section provides a comprehensive answer key for common homework problems involving absolute value and step functions. The problems cover a range of difficulties, from basic evaluations to more complex equation-solving and graphing tasks.

I. Absolute Value Functions

A. Evaluating Absolute Value Expressions

1. Evaluate |7 - 3|:
   • |7 - 3| = |4| = 4
2. Evaluate |-5 + 2|:
   • |-5 + 2| = |-3| = 3
3. Evaluate |3(-4)|:
   • |3(-4)| = |-12| = 12
4. Evaluate |-2| + |5|:
   • |-2| + |5| = 2 + 5 = 7
5. Evaluate |6 - (-2)|:
   • |6 - (-2)| = |6 + 2| = |8| = 8

B. Solving Absolute Value Equations

1. Solve |x| = 5:
   • x = 5 or x = -5
2. Solve |x - 3| = 2:
   • x - 3 = 2 or x - 3 = -2
   • x = 5 or x = 1
3. Solve |2x + 1| = 7:
   • 2x + 1 = 7 or 2x + 1 = -7
   • 2x = 6 or 2x = -8
   • x = 3 or x = -4
4. Solve |4 - x| = 6:
   • 4 - x = 6 or 4 - x = -6
   • -x = 2 or -x = -10
   • x = -2 or x = 10
5. Solve |3x - 5| = 4:
   • 3x - 5 = 4 or 3x - 5 = -4
   • 3x = 9 or 3x = 1
   • x = 3 or x = 1/3

C. Solving Absolute Value Inequalities

1. Solve |x| < 3:
   • -3 < x < 3
2. Solve |x - 2| ≤ 4:
   • -4 ≤ x - 2 ≤ 4
   • -2 ≤ x ≤ 6
3. Solve |2x + 1| < 5:
   • -5 < 2x + 1 < 5
   • -6 < 2x < 4
   • -3 < x < 2
4. Solve |x + 3| > 2:
   • x + 3 > 2 or x + 3 < -2
   • x > -1 or x < -5
5. Solve |3 - x| ≥ 1:
   • 3 - x ≥ 1 or 3 - x ≤ -1
   • -x ≥ -2 or -x ≤ -4
   • x ≤ 2 or x ≥ 4

D. Graphing Absolute Value Functions

1. Graph f(x) = |x + 1|:
   • The vertex is at (-1, 0). The graph is a V-shape opening upwards.
2. Graph f(x) = 2|x| - 3:
   • The vertex is at (0, -3). The graph is a V-shape opening upwards and stretched vertically by a factor of 2.
3. Graph f(x) = -|x - 2| + 1:
   • The vertex is at (2, 1). The graph is a V-shape opening downwards (reflected over the x-axis).
4. Graph f(x) = |2x|:
   • The vertex is at (0, 0). The graph is a V-shape opening upwards and compressed horizontally by a factor of 2.
5. Graph f(x) = |x/3|:
   • The vertex is at (0, 0). The graph is a V-shape opening upwards and stretched horizontally by a factor of 3.

II. Step Functions

A. Evaluating Step Functions

1. Evaluate ⌊4.2⌋:
   • ⌊4.2⌋ = 4
2. Evaluate ⌈-2.8⌉:
   • ⌈-2.8⌉ = -2
3. Evaluate ⌊-5⌋:
   • ⌊-5⌋ = -5
4. Evaluate ⌈3⌉:
   • ⌈3⌉ = 3
5. Evaluate ⌊0.99⌋:
   • ⌊0.99⌋ = 0

B. Solving Equations Involving Step Functions

1. Solve ⌊x⌋ = 3:
   • 3 ≤ x < 4
2. Solve ⌈x⌉ = -2:
   • -3 < x ≤ -2
3. Solve ⌊x/2⌋ = 1:
   • 1 ≤ x/2 < 2
   • 2 ≤ x < 4
4. Solve ⌈2x⌉ = 4:
   • 3 < 2x ≤ 4
   • 1.5 < x ≤ 2
5. Solve ⌊x + 1⌋ = 0:
   • 0 ≤ x + 1 < 1
   • -1 ≤ x < 0

C. Graphing Step Functions

1. Graph f(x) = ⌊x⌋:
   • The graph consists of horizontal line segments at integer values, closed on the left and open on the right.
2. Graph f(x) = ⌈x⌉:
   • The graph consists of horizontal line segments at integer values, open on the left and closed on the right.
3. Graph f(x) = 2⌊x⌋:
   • The graph consists of horizontal line segments at even integer values, closed on the left and open on the right.
4. Graph f(x) = ⌊x - 1⌋:
   • The graph is a horizontal shift of f(x) = ⌊x⌋ one unit to the right.
5. Graph f(x) = ⌈x + 0.5⌉:
   • The graph is a horizontal shift of f(x) = ⌈x⌉ 0.5 units to the left.

D. Applications of Step Functions

1. A phone company charges $0.10 for each minute or part of a minute. Write a function for the cost C(t) for a call lasting t minutes:
   • C(t) = 0.10⌈t⌉
2. A parking garage charges $3 for the first hour and $2 for each additional hour or part of an hour. Write a function for the cost P(h) for parking h hours:
   • P(h) = 3 + 2⌈h - 1⌉, for h > 0; P(h) = 3, for 0 < h ≤ 1
3. The price of stamps is $0.55 for the first ounce and $0.20 for each additional ounce or part of an ounce. Write a function for the cost S(w) for a letter weighing w ounces:
   • S(w) = 0.55 + 0.20⌈w - 1⌉, for w > 0; S(w) = 0.55, for 0 < w ≤ 1
4. An online retailer charges shipping fees based on the total order value. The fees are $5 for orders up to $50, $8 for orders between $50 and $100, and $10 for orders over $100. Write a piecewise function for the shipping cost F(v) for an order value of v dollars:

F(v) = 5, if 0 ≤ v ≤ 50 8, if 50 < v ≤ 100 10, if v > 100

5. A taxi service charges $2.50 for the first mile and $0.50 for each additional mile or part of a mile. Write a function for the cost T(m) for a ride of m miles:
   • T(m) = 2.50 + 0.50⌈m - 1⌉, for m > 0; T(m) = 2.50, for 0 < m ≤ 1

Advanced Problem-Solving Techniques

Combining Absolute Value and Step Functions

Some problems require combining both absolute value and step functions. These can be more challenging but can be approached systematically by breaking down each function and analyzing their combined effects.

1. Graph f(x) = |⌊x⌋|:
   • First, evaluate the floor function, then take the absolute value. The graph will consist of horizontal line segments at non-negative integer values, closed on the left and open on the right.
2. Solve |⌊x⌋| = 2:
   • ⌊x⌋ = 2 or ⌊x⌋ = -2
   • 2 ≤ x < 3 or -2 ≤ x < -1
3. Graph f(x) = ⌊|x|⌋:
   • First, evaluate the absolute value, then take the floor function. The graph will be symmetric about the y-axis, with horizontal line segments at non-negative integer values.
4. Solve ⌊|x|⌋ = 1:
   • 1 ≤ |x| < 2
   • -2 < x ≤ -1 or 1 ≤ x < 2
5. Graph f(x) = ⌈|x|⌉:
   • First, evaluate the absolute value, then take the ceiling function. The graph will be symmetric about the y-axis, with horizontal line segments at non-negative integer values.

Applications in Calculus

Absolute value and step functions are fundamental in calculus, particularly in the study of limits, continuity, and integration.

• Limits: The limit of a function involving absolute values or step functions may not exist at points where the function is discontinuous.
• Continuity: Absolute value functions are continuous everywhere, but step functions are discontinuous at integer values.
• Differentiation: The derivative of an absolute value function exists everywhere except at the point where the expression inside the absolute value is zero. Step functions are not differentiable at the points of discontinuity.
• Integration: Integrals involving absolute value functions require splitting the integral into intervals where the expression inside the absolute value is positive or negative. Integrals involving step functions can be evaluated by summing the areas of the rectangles formed by the steps.

Real-World Examples

1. Tax Brackets: Tax systems often use step functions to define tax brackets, where the tax rate changes at specific income levels.
2. Shipping Costs: Many shipping companies use step functions to determine shipping costs based on weight or distance.
3. Digital Signal Processing: Step functions are used to model digital signals and analyze their behavior in various electronic systems.
4. Control Systems: Engineers use step functions to design control systems that respond to sudden changes in input signals.
5. Actuarial Science: Actuaries use step functions to model insurance payouts and calculate premiums based on various risk factors.

Common Mistakes and How to Avoid Them

1. Incorrectly Applying the Definition of Absolute Value:

   • Mistake: Assuming |x| = x for all x.
   • Solution: Remember that |x| = x if x ≥ 0 and |x| = -x if x < 0.

2. Forgetting to Consider Both Cases in Absolute Value Equations:

   • Mistake: Solving only f(x) = c for |f(x)| = c.
   • Solution: Solve both f(x) = c and f(x) = -c.

3. Incorrectly Interpreting Absolute Value Inequalities:

   • Mistake: Confusing the rules for |f(x)| < c and |f(x)| > c.
   • Solution: Remember that |f(x)| < c implies -c < f(x) < c, while |f(x)| > c implies f(x) < -c or f(x) > c.

4. Misunderstanding the Floor and Ceiling Functions:

   • Mistake: Confusing ⌊x⌋ and ⌈x⌉.
   • Solution: Remember that ⌊x⌋ is the greatest integer less than or equal to x, and ⌈x⌉ is the smallest integer greater than or equal to x.

5. Ignoring the Discontinuities of Step Functions:

   • Mistake: Assuming step functions are continuous everywhere.
   • Solution: Recognize that step functions are discontinuous at integer values (for floor and ceiling functions) or at a specific point (for the Heaviside function).

6. Incorrectly Graphing Transformations of Absolute Value and Step Functions:

   • Mistake: Misinterpreting the effects of a, h, and k in f(x) = a|x - h| + k.
   • Solution: Understand that a stretches or compresses the graph vertically, h shifts the graph horizontally, and k shifts the graph vertically.

Conclusion

Absolute value and step functions are essential mathematical concepts with wide-ranging applications. Understanding their properties, mastering problem-solving techniques, and utilizing resources like answer keys can significantly improve your proficiency in these areas. By avoiding common mistakes and continually practicing, you can build a strong foundation in absolute value and step functions, preparing you for more advanced topics in mathematics and related fields. This detailed answer key and comprehensive explanation should serve as an invaluable tool for your homework and further studies.