Plot the graph, `| f(x)|= log_2(-x)`

Answer

To plot the graph of \( |f(x)| = \log_2(-x) \), we will follow these steps:

### Step 1: Understand the function \( f(x) = \log_2(-x) \)

The function \( \log_2(-x) \) is defined only when \( -x > 0 \), which means \( x < 0 \). Therefore, we will only consider values of \( x \) that are negative.

### Step 2: Plot the graph of \( f(x) = \log_2(-x) \)

1. **Identify the points**:
- When \( x = -1 \), \( f(-1) = \log_2(1) = 0 \).
- When \( x = -2 \), \( f(-2) = \log_2(2) = 1 \).
- When \( x = -4 \), \( f(-4) = \log_2(4) = 2 \).

2. **Sketch the graph**:
- The graph of \( f(x) = \log_2(-x) \) will be in the second quadrant, starting from the point (-1, 0) and increasing as \( x \) approaches 0 from the left.

### Step 3: Reflect the graph for \( |f(x)| \)

Since we need to plot \( |f(x)| \), we will take the portion of the graph of \( f(x) \) that is below the x-axis and reflect it above the x-axis.

1. **Identify the portion below the x-axis**:
- For \( x < -1 \), \( \log_2(-x) \) will be negative.

2. **Reflect the negative part**:
- For \( x < -1 \), if \( f(x) < 0 \), we will take the absolute value, which means reflecting it above the x-axis.

### Step 4: Finalize the graph

1. **Combine the graphs**:
- The graph of \( |f(x)| \) will consist of the original graph for \( x \) values where \( f(x) \) is non-negative (which is only at \( x = -1 \)) and the reflected part for \( x < -1 \).

2. **Sketch the final graph**:
- The graph will start at the point (-1, 0) and will rise as \( x \) approaches 0 from the left, and for \( x < -1 \), it will mirror the values of \( f(x) \) above the x-axis.

### Final Graph

The final graph of \( |f(x)| = \log_2(-x) \) will look like this:

- It will start at the point (-1, 0).
- It will increase without bound as \( x \) approaches 0 from the left.
- For \( x < -1 \), the graph will be a reflection of the values of \( \log_2(-x) \) above the x-axis.