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

Options

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Answer

To solve the problem of finding the correct graph of the function \( y = \frac{|\log_2(2x)|}{\log_2(2x)} \), we will analyze the function step by step.

### Step 1: Simplify the Function
The function can be rewritten as:
\[ y = \frac{|\log_2(2x)|}{\log_2(2x)} \]
This expression depends on the sign of \( \log_2(2x) \).

### Step 2: Determine the Condition for \( \log_2(2x) \)
We need to find when \( \log_2(2x) \) is positive and when it is negative.

1. **When \( \log_2(2x) > 0 \)**:
\[ 2x > 2^0 \implies 2x > 1 \implies x > \frac{1}{2} \]
In this case, \( |\log_2(2x)| = \log_2(2x) \), so:
\[ y = \frac{\log_2(2x)}{\log_2(2x)} = 1 \]

2. **When \( \log_2(2x) < 0 \)**:
\[ 2x < 2^0 \implies 2x < 1 \implies x < \frac{1}{2} \]
Here, \( |\log_2(2x)| = -\log_2(2x) \), so:
\[ y = \frac{-\log_2(2x)}{\log_2(2x)} = -1 \]

### Step 3: Define the Domain
Since \( \log_2(2x) \) is defined only for \( 2x > 0 \), we have:
\[ x > 0 \]
Thus, the function is defined for \( x > 0 \) except at \( x = \frac{1}{2} \) where \( \log_2(2x) = 0 \).

### Step 4: Summary of the Function Behavior
- For \( 0 < x < \frac{1}{2} \), \( y = -1 \).
- For \( x = \frac{1}{2} \), the function is undefined.
- For \( x > \frac{1}{2} \), \( y = 1 \).

### Step 5: Graphing the Function
1. Draw the x-axis and y-axis.
2. For \( 0 < x < \frac{1}{2} \), draw a horizontal line at \( y = -1 \).
3. At \( x = \frac{1}{2} \), there is a hole (undefined point).
4. For \( x > \frac{1}{2} \), draw a horizontal line at \( y = 1 \).

### Final Graph
The graph consists of:
- A horizontal line at \( y = -1 \) for \( 0 < x < \frac{1}{2} \).
- A hole at \( (0.5, -1) \).
- A horizontal line at \( y = 1 \) for \( x > \frac{1}{2} \).