Chapter2: Functions And Their Graphs
Section2.4: A Library Of Parent Functions
Problem 47E: During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate...
Related questions
Question
![### Sketching the Function \( f(x) = \lfloor x + 2 \rfloor - 1 \) on the Interval \([-2, 2]\)
**Problem Statement:**
Sketch \( f(x) = \lfloor x + 2 \rfloor - 1 \) on the interval \([-2, 2]\).
**Function Breakdown:**
This function involves the floor function, denoted as \( \lfloor \cdot \rfloor \), which rounds down to the nearest integer. Specifically, \( f(x) \) can be described in two parts:
1. **Transformation of the Input:**
The expression \( x + 2 \) adjusts the input \( x \) by adding 2. So, every value of \( x \) is increased by 2 before any other operations are applied.
2. **Applying the Floor Function:**
The floor function, \( \lfloor x + 2 \rfloor \), takes the adjusted input and rounds it down to the nearest integer.
3. **Final Adjustment:**
After the floor function has been applied, 1 is subtracted from the result.
**Interval Analysis:**
We consider the function on the range \([-2, 2]\). Here are the steps to generate the sketch:
- **For \( x \in [-2, -1) \):**
\( x + 2 \) ranges from \( 0 \) to \( 1 \). So, \( \lfloor x + 2 \rfloor = \lfloor 0 \leq x + 2 < 1 \rfloor = 0 \).
Therefore, \( f(x) = 0 - 1 = -1 \).
- **For \( x \in [-1, 0) \):**
\( x + 2 \) ranges from \( 1 \) to \( 2 \). So, \( \lfloor 1 \leq x + 2 < 2 \rfloor = 1 \).
Therefore, \( f(x) = 1 - 1 = 0 \).
- **For \( x \in [0, 1) \):**
\( x + 2 \) ranges from \( 2 \) to \( 3 \). So, \( \lfloor 2 \leq x + 2 < 3 \rfloor](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70fe4f99-9e69-4bef-afe6-80d54e8903ee%2Fcbfa86a2-33cf-4b52-b308-578fd28bef8e%2F84j49ku_processed.png&w=3840&q=75)
Transcribed Image Text:### Sketching the Function \( f(x) = \lfloor x + 2 \rfloor - 1 \) on the Interval \([-2, 2]\)
**Problem Statement:**
Sketch \( f(x) = \lfloor x + 2 \rfloor - 1 \) on the interval \([-2, 2]\).
**Function Breakdown:**
This function involves the floor function, denoted as \( \lfloor \cdot \rfloor \), which rounds down to the nearest integer. Specifically, \( f(x) \) can be described in two parts:
1. **Transformation of the Input:**
The expression \( x + 2 \) adjusts the input \( x \) by adding 2. So, every value of \( x \) is increased by 2 before any other operations are applied.
2. **Applying the Floor Function:**
The floor function, \( \lfloor x + 2 \rfloor \), takes the adjusted input and rounds it down to the nearest integer.
3. **Final Adjustment:**
After the floor function has been applied, 1 is subtracted from the result.
**Interval Analysis:**
We consider the function on the range \([-2, 2]\). Here are the steps to generate the sketch:
- **For \( x \in [-2, -1) \):**
\( x + 2 \) ranges from \( 0 \) to \( 1 \). So, \( \lfloor x + 2 \rfloor = \lfloor 0 \leq x + 2 < 1 \rfloor = 0 \).
Therefore, \( f(x) = 0 - 1 = -1 \).
- **For \( x \in [-1, 0) \):**
\( x + 2 \) ranges from \( 1 \) to \( 2 \). So, \( \lfloor 1 \leq x + 2 < 2 \rfloor = 1 \).
Therefore, \( f(x) = 1 - 1 = 0 \).
- **For \( x \in [0, 1) \):**
\( x + 2 \) ranges from \( 2 \) to \( 3 \). So, \( \lfloor 2 \leq x + 2 < 3 \rfloor
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Intermediate Algebra
Algebra
ISBN:
9781285195728
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Intermediate Algebra
Algebra
ISBN:
9781285195728
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill