Of course. Here is a complete, in-depth educational article on Absolute Value and Step Functions, structured as a comprehensive homework answer key Not complicated — just consistent..
Mastering Absolute Value and Step Functions: Your Complete Homework Answer Key
Navigating the world of algebra can often feel like decoding a new language. Among the most important "sentences" in this language are absolute value and step functions. These concepts are fundamental, appearing in everything from computer programming and physics to economics and data science. If you're tackling a homework assignment on these topics, you've come to the right place. This guide will not only provide the answers you need but, more importantly, break down the why and how behind each solution, ensuring you build a solid understanding that will serve you well beyond this assignment Small thing, real impact..
No fluff here — just what actually works The details matter here..
This article is designed to be your ultimate resource. We will explore the core concepts, walk through detailed, step-by-step solutions for common homework problems, and clarify the underlying mathematical principles. Think of it as your personal tutor, available 24/7.
H2: Part 1: The Absolute Value – More Than Just Making Things Positive
The absolute value of a number is its distance from zero on the number line. On the flip side, because distance is always non-negative, the absolute value is never negative. We denote the absolute value of a number x as |x|.
- The Definition: |x| = x if x ≥ 0, and |x| = -x if x < 0.
This simple definition is the key to solving any absolute value equation or inequality. The critical insight is that an absolute value equation |A| = B (where B ≥ 0) actually represents two separate equations: A = B or A = -B That's the part that actually makes a difference..
Common Homework Problems & Step-by-Step Solutions
Problem 1: Solve the equation |2x - 5| = 9.
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Step 1: Set up the two separate equations based on the definition Simple, but easy to overlook..
- Case 1: 2x - 5 = 9
- Case 2: 2x - 5 = -9
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Step 2: Solve each equation individually.
- For Case 1: Add 5 to both sides: 2x = 14. Divide by 2: x = 7.
- For Case 2: Add 5 to both sides: 2x = -4. Divide by 2: x = -2.
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Step 3: Check your solutions No workaround needed..
- For x = 7: |2(7) - 5| = |14 - 5| = |9| = 9. Correct.
- For x = -2: |2(-2) - 5| = |-4 - 5| = |-9| = 9. Correct.
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Answer: The solutions are x = 7 and x = -2 And that's really what it comes down to..
Problem 2: Solve the inequality |x + 3| ≤ 7.
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Step 1: Translate the absolute value inequality into a compound inequality.
- The inequality |A| ≤ B (where B ≥ 0) is equivalent to -B ≤ A ≤ B.
- So, -7 ≤ x + 3 ≤ 7.
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Step 2: Isolate the variable x by subtracting 3 from all three parts of the inequality Small thing, real impact..
- -7 - 3 ≤ x ≤ 7 - 3
- -10 ≤ x ≤ 4
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Answer: The solution is the interval [-10, 4]. In set-builder notation: {x | -10 ≤ x ≤ 4}.
Problem 3: Graph the function f(x) = |x - 2| + 1.
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Step 1: Identify the vertex. The standard form is f(x) = |x - h| + k, where (h, k) is the vertex. Here, h = 2 and k = 1, so the vertex is at (2, 1).
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Step 2: Determine the shape. The coefficient of the absolute value is positive 1, so the graph is a "V" shape that opens upwards Easy to understand, harder to ignore..
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Step 3: Find additional points. Choose x-values around the vertex.
- For x = 3: f(3) = |3 - 2| + 1 = 1 + 1 = 2. Point: (3, 2)
- For x = 1: f(1) = |1 - 2| + 1 = 1 + 1 = 2. Point: (1, 2)
- For x = 4: f(4) = |4 - 2| + 1 = 2 + 1 = 3. Point: (4, 3)
- For x = 0: f(0) = |0 - 2| + 1 = 2 + 1 = 3. Point: (0, 3)
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Answer: Plot the vertex (2,1) and the symmetric points (3,2), (1,2), (4,3), and (0,3). Connect them to form a "V" with the vertex at the bottom That's the part that actually makes a difference..
H2: Part 2: Step Functions – Graphs That Look Like Stairs
A step function is a piecewise constant function that changes its value at specific points, creating a graph that resembles a series of steps. The most common example is the Greatest Integer Function, also known as the floor function, denoted as f(x) = ⌊x⌋. This function outputs the greatest integer less than or equal to x.
- Examples:
- ⌊3.7⌋ = 3
- ⌊-1.2⌋ = -2 (This is a common point of confusion! The greatest integer less than -1.2 is -2.)
- ⌊5⌋ = 5
Common Homework Problems & Step-by-Step Solutions
Problem 4: Evaluate the step function f(x) = ⌊x⌋ for the given values.
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a) f(4.9)
- The greatest integer less than or equal to 4.9 is 4.
- Answer: 4
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b) f(-3.1)
- The greatest integer less than or equal to -3.1 is -4.
- Answer: -4
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c) f(0)
- The greatest integer less than or equal to 0 is 0.
- Answer: 0
Problem 5: Graph the function g(x) = ⌊x⌋ + 2.
- Step 1: Understand the base function. The graph of ⌊x⌋ consists of horizontal line segments.
- For any x in the interval [0, 1), ⌊x⌋ = 0.
- For any x in the interval [1, 2), ⌊x⌋ = 1.
- For any x in the interval [-1, 0), ⌊x⌋ = -1.
- Each segment is closed on the left (solid dot) and open on the right (open circle).
