Sketch the graph of f. fx) 5 fx) 5 4 3 2 2 1 1 3 4 4 -3 -2 -1 1 3 4 -4 -3 -2 -1 -1 -1 f(x) f(x) 5, 3 3 2 2 1. X 3 4 -4 -3 -2 -1 2 3 4 -4 -3 -2 -1 -1

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Sketch the Graph of \( f \)

#### Introduction
In this activity, you are asked to choose the correct graph that represents a function \( f(x) \). Each graph is plotted with the x-axis ranging from -4 to 4 and the y-axis ranging from -1 to 5. Take note of the behavior of the function at key points to determine which graph matches the given description.

#### Graph Descriptions

1. **Graph (i)**:
   - **x-axis**: From -4 to 4.
   - **y-axis**: From -1 to 5.
   - **Key Points**:
     - The function starts at \( f(x) = -1 \) when \( x = -4 \).
     - The function increases to a maximum point at \( f(x) = 3 \) when \( x \approx 1 \).
     - The function then decreases to \( f(x) = 0 \) when \( x = 4 \).

2. **Graph (ii)**:
   - **x-axis**: From -4 to 4.
   - **y-axis**: From -1 to 5.
   - **Key Points**:
     - The function starts at \( f(x) = -1 \) when \( x = -4 \).
     - The function increases sharply reaching a peak at \( f(x) = 4 \) when \( x \approx -1 \).
     - There is another peak at \( f(x) = 3\) when \( x \approx 1 \).
     - The function then decreases to \( f(x) = 0 \) when \( x = 4 \).

3. **Graph (iii)**:
   - **x-axis**: From -4 to 4.
   - **y-axis**: From -1 to 5.
   - **Key Points**:
     - The function starts at \( f(x) = -1 \) when \( x = -4 \).
     - The function increases steadily reaching a peak at \( f(x) = 4 \) when \( x \approx -2 \).
     - Another peak is reached at \( f(x) = 3\) when \( x \approx 1 \).
     - The function then decreases to \( f(x) = 1 \) when \( x = 4 \).

4. **Graph (iv)**:
   -
Transcribed Image Text:### Sketch the Graph of \( f \) #### Introduction In this activity, you are asked to choose the correct graph that represents a function \( f(x) \). Each graph is plotted with the x-axis ranging from -4 to 4 and the y-axis ranging from -1 to 5. Take note of the behavior of the function at key points to determine which graph matches the given description. #### Graph Descriptions 1. **Graph (i)**: - **x-axis**: From -4 to 4. - **y-axis**: From -1 to 5. - **Key Points**: - The function starts at \( f(x) = -1 \) when \( x = -4 \). - The function increases to a maximum point at \( f(x) = 3 \) when \( x \approx 1 \). - The function then decreases to \( f(x) = 0 \) when \( x = 4 \). 2. **Graph (ii)**: - **x-axis**: From -4 to 4. - **y-axis**: From -1 to 5. - **Key Points**: - The function starts at \( f(x) = -1 \) when \( x = -4 \). - The function increases sharply reaching a peak at \( f(x) = 4 \) when \( x \approx -1 \). - There is another peak at \( f(x) = 3\) when \( x \approx 1 \). - The function then decreases to \( f(x) = 0 \) when \( x = 4 \). 3. **Graph (iii)**: - **x-axis**: From -4 to 4. - **y-axis**: From -1 to 5. - **Key Points**: - The function starts at \( f(x) = -1 \) when \( x = -4 \). - The function increases steadily reaching a peak at \( f(x) = 4 \) when \( x \approx -2 \). - Another peak is reached at \( f(x) = 3\) when \( x \approx 1 \). - The function then decreases to \( f(x) = 1 \) when \( x = 4 \). 4. **Graph (iv)**: -
**Finding Discontinuities and Continuity Types**

This section covers the topic of discontinuities in piecewise functions and how to determine the type of continuity at each point of discontinuity.

### Problem Statement

Given a piecewise function \( f(x) \):

\[ f(x) = \begin{cases} 
x + 4 & \text{if } x < 0 \\
4x^2 & \text{if } 0 \leq x \leq 1 \\
4 - x & \text{if } x > 1 
\end{cases} \]

Your tasks are:
1. Find each \( x \)-value at which \( f \) is discontinuous.
2. For each \( x \)-value, determine whether \( f \) is continuous from the right, from the left, or neither.

#### Steps and Questions:

1. Identify the smaller \( x \)-value for discontinuities.

   \( x = \_\_\_\_\_\_ \)

   - [ ] Continuous from the right
   - [ ] Continuous from the left
   - [ ] Neither

2. Identify the larger \( x \)-value for discontinuities.

   \( x = \_\_\_\_\_\_ \)

   - [ ] Continuous from the right
   - [ ] Continuous from the left
   - [ ] Neither

3. Sketch the graph of \( f \).

---

### Detailed Analysis

#### Step 1: Checking for Discontinuities

To find the discontinuities, observe the transition points between the different pieces of the piecewise function. That is where \( x = 0 \) and \( x = 1 \).

#### Step 2: Left and Right Continuity

For \( x = 0 \):

- From the left (\( x \to 0^{-} \)):
  \[ \lim_{{x \to 0^{-}}}(x + 4) = 0 + 4 = 4 \]
- From the right (\( x \to 0^{+} \)):
  \[ \lim_{{x \to 0^{+}}}(4x^2) = 4(0)^2 = 0 \]

Clearly, these limits are not equal; hence, \( f \) is not continuous at \( x = 0 \). 

For \( x = 1 \):

- From the left (\(
Transcribed Image Text:**Finding Discontinuities and Continuity Types** This section covers the topic of discontinuities in piecewise functions and how to determine the type of continuity at each point of discontinuity. ### Problem Statement Given a piecewise function \( f(x) \): \[ f(x) = \begin{cases} x + 4 & \text{if } x < 0 \\ 4x^2 & \text{if } 0 \leq x \leq 1 \\ 4 - x & \text{if } x > 1 \end{cases} \] Your tasks are: 1. Find each \( x \)-value at which \( f \) is discontinuous. 2. For each \( x \)-value, determine whether \( f \) is continuous from the right, from the left, or neither. #### Steps and Questions: 1. Identify the smaller \( x \)-value for discontinuities. \( x = \_\_\_\_\_\_ \) - [ ] Continuous from the right - [ ] Continuous from the left - [ ] Neither 2. Identify the larger \( x \)-value for discontinuities. \( x = \_\_\_\_\_\_ \) - [ ] Continuous from the right - [ ] Continuous from the left - [ ] Neither 3. Sketch the graph of \( f \). --- ### Detailed Analysis #### Step 1: Checking for Discontinuities To find the discontinuities, observe the transition points between the different pieces of the piecewise function. That is where \( x = 0 \) and \( x = 1 \). #### Step 2: Left and Right Continuity For \( x = 0 \): - From the left (\( x \to 0^{-} \)): \[ \lim_{{x \to 0^{-}}}(x + 4) = 0 + 4 = 4 \] - From the right (\( x \to 0^{+} \)): \[ \lim_{{x \to 0^{+}}}(4x^2) = 4(0)^2 = 0 \] Clearly, these limits are not equal; hence, \( f \) is not continuous at \( x = 0 \). For \( x = 1 \): - From the left (\(
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