Absolute Value And Step Functions Homework Answer Key

Absolute Value and Step Functions: Homework Answer Key

Absolute value and step functions are fundamental concepts in mathematics that appear in various branches from algebra to calculus. Understanding these functions is crucial for solving equations, graphing, and analyzing real-world situations. This practical guide will help you master these concepts with clear explanations, examples, and solutions to common homework problems The details matter here..

Understanding Absolute Value Functions

The absolute value of a number represents its distance from zero on the number line, regardless of direction. Mathematically, the absolute value function is denoted as |x| and defined as:

|x| = x, if x ≥ 0 |x| = -x, if x < 0

This piecewise definition means that for non-negative numbers, the absolute value is the number itself, while for negative numbers, it's the opposite (positive) value.

Properties of Absolute Value

1. Non-negativity: |x| ≥ 0 for all real numbers x
2. Identity: |x| = 0 if and only if x = 0
3. Symmetry: |-x| = |x| for all real numbers x
4. Triangle inequality: |x + y| ≤ |x| + |y| for all real numbers x and y
5. Multiplication property: |xy| = |x|·|y| for all real numbers x and y

Graphing Absolute Value Functions

The graph of y = |x| is a V-shaped graph with the vertex at the origin (0,0). For more complex absolute value functions like y = |x - h| + k, the vertex shifts to (h,k) And it works..

Example: Graph y = |x - 3| + 2

1. Identify the vertex: (3,2)
2. The graph opens upward (since there's no negative coefficient)
3. Choose points on both sides of x = 3:
   • When x = 1: y = |1 - 3| + 2 = 4
   • When x = 5: y = |5 - 3| + 2 = 4
   • When x = 3: y = |3 - 3| + 2 = 2

Understanding Step Functions

Step functions are piecewise functions that are constant over specific intervals and change abruptly at certain points. The most common step function is the greatest integer function, also known as the floor function, denoted as ⌊x⌋, which gives the greatest integer less than or equal to x.

Types of Step Functions

1. Floor function: ⌊x⌋ = greatest integer ≤ x
2. Ceiling function: ⌈x⌉ = smallest integer ≥ x
3. Unit step function: u(x) = 0 if x < 0, 1 if x ≥ 0
4. Characteristic function: χ_A(x) = 1 if x ∈ A, 0 otherwise

Graphing Step Functions

Step functions appear as "steps" or "staircases" on a graph. To give you an idea, the floor function ⌊x⌋ will have horizontal line segments at integer values, with jumps at each integer.

Example: Graph y = ⌊x⌋ for -3 ≤ x ≤ 3

1. For -3 ≤ x < -2: y = -3
2. For -2 ≤ x < -1: y = -2
3. For -1 ≤ x < 0: y = -1
4. For 0 ≤ x < 1: y = 0
5. For 1 ≤ x < 2: y = 1
6. For 2 ≤ x ≤ 3: y = 2

Common Homework Problems and Solutions

Absolute Value Equations

Problem: Solve |2x - 5| = 7

Solution: When solving absolute value equations, we consider two cases:

Case 1: 2x - 5 = 7 2x = 12 x = 6

Case 2: 2x - 5 = -7 2x = -2 x = -1

Which means, the solutions are x = 6 and x = -1.

Absolute Value Inequalities

Problem: Solve |3x + 2| < 10

Solution: For |A| < B (where B > 0), we have -B < A < B

-10 < 3x + 2 < 10 -12 < 3x < 8 -4 < x < 8/3

That's why, the solution is -4 < x < 8/3 It's one of those things that adds up. Nothing fancy..

Step Function Applications

Problem: A parking lot charges $5 for the first hour or part thereof, and $3 for each additional hour or part thereof. Write a function for the cost C(t) for t hours of parking.

Solution: This is a step function where the cost increases after each hour:

C(t) = 5, if 0 < t ≤ 1 C(t) = 8, if 1 < t ≤ 2 C(t) = 11, if 2 < t ≤ 3 ...

In general form: C(t) = 5 + 3⌊t⌋ for t > 0, where ⌊t⌋ is the greatest integer less than or equal to t.

Composite Functions with Absolute Value

Problem: If f(x) = |x + 3| and g(x) = 2x - 1, find f(g(x)) and g(f(x))

Solution:

f(g(x)) = f(2x - 1) = |(2x - 1) + 3| = |2x + 2|

g(f(x)) = g(|x + 3|) = 2|x + 3| - 1

Practice Problems with Solutions

Problem 1: Solve |4x - 3| ≥ 5

Solution: For |A| ≥ B (where B > 0), we have A ≤ -B or A ≥ B

Case 1: 4x - 3 ≥ 5 4x ≥ 8 x ≥ 2

Case 2: 4x - 3 ≤ -5 4x ≤ -2 x ≤ -0.5

Which means, the solution is x ≤ -0.5 or x ≥ 2.

Problem 2: Graph the function y = 2⌊x⌋ - 1 for -2 ≤ x ≤ 3

Solution: This is a transformation of the floor function:

1. For -2 ≤ x < -1: y = 2(-2) - 1 = -5
2. For -1 ≤ x < 0: y = 2(-1) - 1 = -3
3. For 0 ≤ x < 1: y = 2(0) - 1 = -1
4. For 1 ≤ x < 2: y = 2(1) - 1 = 1
5. For

Problem 3: Solve ( |5x + 2| \leq 9 )

Solution

For (|A| \le B) we have (-B \le A \le B) Took long enough..

[ -9 \le 5x + 2 \le 9 ]

Subtract 2:

[ -11 \le 5x \le 7 ]

Divide by 5:

[ -\frac{11}{5} \le x \le \frac{7}{5} ]

So the solution set is (x \in \left[-\frac{11}{5},, \frac{7}{5}\right]) Not complicated — just consistent..

Problem 4: Model a Toll Booth Charge

A toll booth charges $2 for each full mile traveled and an additional $1 if any part of a mile is driven beyond a full mile. Write a step‑function model for the toll (T(d)) in dollars as a function of distance (d) in miles ((d \ge 0)) It's one of those things that adds up..

Solution

The function jumps whenever a new mile is entered. It can be expressed as

[ T(d) = 2,\lceil d \rceil ]

where (\lceil d \rceil) is the ceiling function (the smallest integer (\ge d)).
For example:

Putting It All Together: A Real‑World Scenario

Suppose a city parking meter charges $4 for the first 30 minutes and $2 for each subsequent 30‑minute block (or part thereof). We want to write a function (P(t)) that gives the total cost in dollars for (t) minutes of parking.

1. First 30 minutes: $4 (always charged)
2. Remaining time: (\max(0, t-30)) minutes
3. Number of 30‑minute blocks in the remaining time: (\left\lceil \frac{\max(0, t-30)}{30} \right\rceil)
4. Cost for extra blocks: (2 \times \left\lceil \frac{\max(0, t-30)}{30} \right\rceil)

Putting it together:

[ P(t) = 4 + 2 \left\lceil \frac{\max(0,, t-30)}{30} \right\rceil, \qquad t \ge 0 ]

This is a classic step function: the cost jumps every 30 minutes after the first block Most people skip this — try not to. That's the whole idea..

Key Takeaways

1. Step functions are piecewise constant functions that remain flat over intervals and jump at specific points.
2. The floor (\lfloor x \rfloor) and ceiling (\lceil x \rceil) functions are the most common building blocks for constructing step functions.
3. When solving absolute value equations or inequalities, always split into cases based on the sign of the expression inside the absolute value.
4. Composite functions involving absolute value or step functions are handled by applying the inner function first, then the outer one, respecting the domain restrictions.
5. Real‑world pricing schemes (parking, tolls, utilities) naturally lead to step‑function models, making them a powerful tool for both mathematics education and practical analysis.

Final Words

Step functions may look simple—just horizontal segments and sudden jumps—but they encapsulate a wealth of mathematical ideas: integer rounding, piecewise definition, and discrete modeling. Mastering them equips you with a versatile toolkit for tackling problems in algebra, calculus, and applied contexts alike. Whether you’re graphing the floor function, solving an absolute value inequality, or designing a pricing scheme, remember that the essence of a step function is its constancy over intervals and immediacy of change at the boundaries. With this understanding, you can confidently manage both the theoretical and the practical landscapes where step functions play a starring role The details matter here..