Which of following functions have the same graph?

Options

A. `f(x)=log_(e)e^(x)`
B. `g(x)=abs(x)sgnx`
C. `h(x)=cot^(-1)(cotx)`
D. `k(x)=lim_(n rarr infty)(2abs(x))/pi tan^(-1)(nx)`

Answer

To determine which of the given functions have the same graph, we will analyze each function step by step.

### Step 1: Analyze the first function \( f(x) = \log_e(e^x) \)

1. **Domain**: The logarithm function requires its argument to be positive. Since \( e^x \) is always positive for all \( x \), the domain of \( f(x) \) is \( x \in \mathbb{R} \).
2. **Simplification**: By properties of logarithms, we can simplify:
\[ f(x) = \log_e(e^x) = x \]
3. **Graph**: The graph of \( f(x) = x \) is a straight line passing through the origin with a slope of 1.

### Step 2: Analyze the second function \( g(x) = | \text{sgn}(x) | x \)

1. **Definition of sgn**: The signum function \( \text{sgn}(x) \) is defined as:
- \( 1 \) when \( x > 0 \)
- \( 0 \) when \( x = 0 \)
- \( -1 \) when \( x < 0 \)
2. **Cases**:
- When \( x > 0 \): \( g(x) = |1| x = x \)
- When \( x = 0 \): \( g(x) = |0| \cdot 0 = 0 \)
- When \( x < 0 \): \( g(x) = |-1| x = -x \) (but since we take absolute value, it becomes \( -(-x) = x \))
3. **Graph**: Thus, \( g(x) = x \) for all \( x \). The graph is also a straight line passing through the origin with a slope of 1.

### Step 3: Analyze the third function \( h(x) = \cot^{-1}(\cot(x)) \)

1. **Understanding the function**: The function \( \cot^{-1}(\cot(x)) \) is defined for \( x \) in the intervals where \( \cot(x) \) is defined, but it is discontinuous at certain points.
2. **Graph**: The graph of \( h(x) \) does not match the linear graph of \( f(x) \) and \( g(x) \). It has discontinuities and is not a straight line.

### Step 4: Analyze the fourth function \( k(x) = \frac{2x}{\pi} \) for \( x > 0 \) and \( -\frac{2x}{\pi} \) for \( x < 0 \)

1. **Cases**:
- When \( x > 0 \): \( k(x) = \frac{2x}{\pi} \) (as \( n \to \infty \), this approaches \( x \))
- When \( x < 0 \): \( k(x) = -\frac{2x}{\pi} \) (as \( n \to \infty \), this also approaches \( x \))
- When \( x = 0 \): \( k(0) = 0 \)
2. **Graph**: Thus, \( k(x) = x \) for all \( x \). The graph is also a straight line passing through the origin with a slope of 1.

### Conclusion

From the analysis:
- The functions \( f(x) \), \( g(x) \), and \( k(x) \) all simplify to \( x \) and have the same graph (a straight line).
- The function \( h(x) \) does not match the others.

### Final Answer

The functions that have the same graph are:
- \( f(x) = \log_e(e^x) \)
- \( g(x) = | \text{sgn}(x) | x \)
- \( k(x) = \frac{2x}{\pi} \) (for \( x > 0 \) and \( x < 0 \))