Let's solve the problem step-by-step: ### **Step 1: Understanding the Functions** 1. **\( y = \sin x \) (blue curve)** - This is the standard sine function. - **Domain**: \( x \in (-\infty, \infty) \) - **Range**: \( y \in [-1, 1] \) 2. **\( y = \lceil \sin x \rceil \) (red curve)** - This is the **ceiling function** applied to the sine function. - The ceiling function, \( \lceil y \rceil \), gives the smallest integer greater than or equal to \( y \). - For \( \sin x \), which ranges from -1 to 1, \( \lceil \sin x \rceil \) can take the values \(-1, 0, 1\): - When \( -1 \leq \sin x < 0 \), \( \lceil \sin x \rceil = 0 \) - When \( \sin x = 0 \), \( \lceil \sin x \rceil = 0 \) - When \( 0 < \sin x < 1 \), \( \lceil \sin x \rceil = 1 \) - When \( \sin x = 1 \), \( \lceil \sin x \rceil = 1 \) - When \( \sin x = -1 \), \( \lceil \sin x \rceil = -1 \) --- ### **Step 2: Domain and Range of \( \lceil \sin x \rceil \)** - **Domain:** \( (-\infty, \infty) \) (same as \(\sin x\)) - **Range:** \( \{-1, 0, 1\} \) --- ### **Step 3: Identifying the Correct Graph** - The blue curve should be a smooth sine wave. - The red curve should be a step function that jumps between \(-1\), \(0\), and \(1\): - At \( \sin x = -1 \): red at \(-1\) - For \( -1 < \sin x < 0 \): red at \(0\) - For \( 0 < \sin x < 1 \): red at \(1\) - At \( \sin x = 0 \): red at \(0\) - At \( \sin x = 1 \): red at \(1\) #### **Analyzing the Choices:** - **Graph A:** The red step function jumps between 0 and 1, but does *not* go down to -1. - **Graph B:** The red step function jumps between -1, 0, and 1. This matches our expectation. - **Graph C:** The red step function jumps between 1 and 0, but appears to be shifted. - **Graph D:** The red step function jumps between 1 and 0, but appears to be shifted as well. **Therefore, the correct answer is:** --- ### **Final Answer** **Correct graph: B** --- **Domain of \( \lceil \sin x \rceil \):** \( (-\infty, \infty) \) **Range of \( \lceil \sin x \rceil \):** \( \{-1, 0, 1\} \)

Let's analyze each option carefully, explaining how each graph corresponds to the functions, and then identify the correct one. --- ### **Step 1: Understanding the functions** - **\( y = \sin x \)** - Is a smooth, periodic wave oscillating between \(-1\) and \(1\). - **Domain:** All real numbers, \( (-\infty, \infty) \) - **Range:** \( [-1, 1] \) - **\( y = \lceil \sin x \rceil \)** - Is the ceiling of \( \sin x \). - Since \( \sin x \in [-1, 1] \), the ceiling function will take values: - \(-1\) when \( \sin x = -1 \), - \(0\) when \( -1 < \sin x \leq 0 \), - \(1\) when \( 0 < \sin x \leq 1 \), - At the endpoints, the ceiling jumps accordingly. - **Range of \( \lceil \sin x \rceil \):** \( \{-1, 0, 1\} \) --- ### **Step 2: Expected graph features** - The graph of \( y = \sin x \) is a smooth wave. - The graph of \( y = \lceil \sin x \rceil \) will be a step function: - **At points where \( \sin x = -1 \):** \( y = -1 \), - **When \( \sin x \) is between \(-1\) and \(0\):** \( y = 0 \), - **When \( \sin x \) is between \(0\) and \(1\):** \( y = 1 \), - **At \( \sin x = 0 \):** \( y = 0 \), - **At \( \sin x = 1 \):** \( y=1 \). The red step function should have constant segments at these three values, jumping at points where \( \sin x \) crosses the thresholds. --- ### **Step 3: Analyzing the options** #### **Option A:** - The red graph appears as a step that only takes values 0 and 1. - It **does not** go down to \(-1\). - **Incorrect** because \( \lceil \sin x \rceil \) should also be \(-1\) at points where \( \sin x = -1 \). --- #### **Option B:** - The red graph jumps between \(-1\), \(0\), and \(1\). - The step levels match the expected values: - At \( \sin x = -1 \), red at \(-1\). - When \( -1 < \sin x < 0 \), red at 0. - When \( 0 < \sin x < 1 \), red at 1. - At \( \sin x=0 \), red at 0. - At \( \sin x=1 \), red at 1. - The red steps align with the sine wave crossings. - **Correct**. --- #### **Option C:** - The red steps appear shifted and only switch between 0 and 1. - It **misses** the \(-1\) level. - **Incorrect**. --- #### **Option D:** - Similar to C, but the steps are positioned differently. - Also missing the \(-1\) level. - **Incorrect**. --- ### **Final conclusion:** The only graph that correctly displays the ceiling of the sine function with steps at \(-1\), 0, and 1 is **Option B**. --- ### **Final answer:** **The correct graph is:** **Option B** --- ### **Summary of domain and range of \( y = \lceil \sin x \rceil \):** - **Domain:** \( (-\infty, \infty) \) (all real numbers) - **Range:** \( \{-1, 0, 1\} \) --- **Thus, the correct choice is: B.**