Absolute Value And Step Functions Homework Answer Key

Absolute Value and Step Functions: A Homework Answer Key and Deep Dive

Struggling with absolute value and step function problems is a common rite of passage in algebra and pre-calculus. It’s one thing to understand the definitions, but applying them to complex homework questions can feel like navigating a maze. This guide serves as more than just an answer key; it’s a comprehensive walkthrough designed to build your conceptual understanding and problem-solving skills. We will dissect the core principles, explore their powerful intersection, and provide detailed, step-by-step solutions to typical homework problems, transforming confusion into clarity.

Understanding the Foundations: Absolute Value and the Step Function

Before tackling combined problems, a solid grasp of each function individually is non-negotiable.

The Absolute Value Function: f(x) = |x| At its heart, the absolute value represents distance from zero on the number line. This means it is always non-negative. The piecewise definition is crucial: |x| = x if x ≥ 0 |x| = -x if x < 0 Graphically, this creates the iconic "V-shape" with its vertex at the origin (0,0). For |x - h|, the vertex shifts to (h, 0). The expression inside the bars dictates the location of this vertex. When solving equations like |expression| = k, you must consider two cases: expression = k and expression = -k, provided k ≥ 0.

The Step Function (Greatest Integer Function): f(x) = [[x]] Also known as the floor function, [[x]] gives the greatest integer less than or equal to x. For any real number, you "step down" to the nearest integer. Its graph is a series of horizontal line segments, each with a closed circle on the left (inclusive) and an open circle on the right (exclusive), creating a staircase pattern. For example, [[3.7]] = 3, [[-1.2]] = -2 (not -1, because -2 is less than -1.2 and is the greatest integer that satisfies the condition). Functions like [[x]] + 1 or [[2x]] modify the step height and width.

The Powerful Intersection: Why Combine Them?

Homework problems often combine these functions to test your ability to analyze piecewise behavior and discontinuities. A typical form is f(x) = | [[x]] | or g(x) = [[ |x| ]]. These compositions force you to apply the inner function first, then the outer, meticulously tracking how the output changes across intervals. The resulting graph is a fascinating hybrid—a staircase whose steps are either all positive or mirrored, depending on the order of operations. Mastering this intersection is key to solving advanced piecewise and transformation problems.

Sample Homework Problems and Detailed Answer Key

Here are representative problems you might encounter, solved with full reasoning.

Problem 1: Graphing and Analysis Sketch the graph of f(x) = | [[x]] | for -3 ≤ x ≤ 3. Identify its range and any points of discontinuity.

Step-by-Step Solution:

1. Evaluate the Inner Function First: We must compute [[x]] for intervals of length 1.
   • For -3 ≤ x < -2: [[x]] = -3 → | -3 | = 3
   • For -2 ≤ x < -1: [[x]] = -2 → | -2 | = 2
   • For -1 ≤ x < 0: [[x]] = -1 → | -1 | = 1
   • For 0 ≤ x < 1: [[x]] = 0 → | 0 | = 0
   • For 1 ≤ x < 2: [[x]] = 1 → | 1 | = 1
   • For 2 ≤ x < 3: [[x]] = 2 → | 2 | = 2
   • At x = 3: [[3]] = 3 → | 3 | = 3
2. Plot the Horizontal Segments: On each interval, f(x) is constant.
   • A horizontal line at y=3 from x=-3 (closed) to x=-2 (open).
   • A horizontal line at y=2 from x=-2 (closed) to x=-1 (open).
   • A horizontal line at y=1 from x=-1 (closed) to x=0 (open).
   • A horizontal line at y=0 from x=0 (closed) to x=1 (open).
   • A horizontal line at y=1 from x=1 (closed) to x=2 (open).
   • A horizontal line at y=2 from x=2 (closed) to x=3 (open).
   • A single point at (3, 3).
3. Analyze:
   • Range: The set of all y-values is {0, 1, 2, 3}.
   • Discontinuities (Jumps): The function jumps at every integer x = -2, -1, 0, 1, 2. At x=0, the jump is from y=1 (left) to y=0 (right). At all other integers listed, the jump is from a lower y-value to a higher one (e.g., at x=-2, from y=3 to y=2).

Problem 2: Solving an Equation Solve for x: | [[x]] | = 1.5

Step-by-Step Solution:

1. Interpret the Equation: The output of | [[x]] | must equal 1.5. However, [[x]] is always an integer. The absolute value of an integer is a non-negative integer (0, 1, 2, 3...).
2. Identify the Contradiction: 1.5 is not an integer. There is no integer n such that `|n| =