The correct graph of `y = (|log_(2)x|)/(log_(2)x)` is

Options

A.
B.
C.
D.

Explanation

To solve the problem of graphing the function \( y = \frac{|\log_2 x|}{\log_2 x} \), we will analyze the function step by step.

### Step 1: Understanding the Function
The function is defined as:
\[ y = \frac{|\log_2 x|}{\log_2 x} \]

### Step 2: Determine the Domain
The logarithm function \( \log_2 x \) is defined for \( x > 0 \). Therefore, the domain of the function is \( x > 0 \).

### Step 3: Analyze the Sign of \( \log_2 x \)
1. **For \( x > 1 \)**:
- \( \log_2 x > 0 \)
- Thus, \( |\log_2 x| = \log_2 x \)
- Therefore:
\[ y = \frac{\log_2 x}{\log_2 x} = 1 \]

2. **For \( 0 < x < 1 \)**:
- \( \log_2 x < 0 \)
- Thus, \( |\log_2 x| = -\log_2 x \)
- Therefore:
\[ y = \frac{-\log_2 x}{\log_2 x} = -1 \]

### Step 4: Summary of Values
- For \( x > 1 \), \( y = 1 \)
- For \( 0 < x < 1 \), \( y = -1 \)

### Step 5: Graph the Function
- The graph will have a horizontal line at \( y = 1 \) for \( x > 1 \).
- The graph will have a horizontal line at \( y = -1 \) for \( 0 < x < 1 \).
- At \( x = 1 \), \( \log_2 1 = 0 \) is not defined in the function, so there is a discontinuity at this point.

### Step 6: Final Graph
The graph consists of two horizontal lines:
- A line at \( y = 1 \) for \( x > 1 \)
- A line at \( y = -1 \) for \( 0 < x < 1 \)
- There is a discontinuity at \( x = 1 \).

### Conclusion
The correct graph of \( y = \frac{|\log_2 x|}{\log_2 x} \) is a piecewise function that takes the value of 1 for \( x > 1 \) and -1 for \( 0 < x < 1 \).

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