The number of solutions to `|e^(|x|)-3|=K` is

Options

A. Two if `K=0`
B. Three if `Kgt2`
C. Three if `K=2`
D. Four if `K=1`

Explanation

To solve the equation \( |e^{|x|} - 3| = K \), we will analyze the function and determine the number of solutions based on the value of \( K \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The equation can be rewritten as two separate cases due to the absolute value:
\[ e^{|x|} - 3 = K \quad \text{or} \quad e^{|x|} - 3 = -K \]

2. **Finding the Function Behavior**:
The function \( e^{|x|} \) is always positive and increases as \( |x| \) increases. Thus, we can analyze the two cases separately.

3. **Case 1: \( e^{|x|} - 3 = K \)**:
Rearranging gives:
\[ e^{|x|} = K + 3 \]
For this equation to have solutions, \( K + 3 \) must be greater than 0, which means \( K > -3 \).

4. **Case 2: \( e^{|x|} - 3 = -K \)**:
Rearranging gives:
\[ e^{|x|} = 3 - K \]
For this equation to have solutions, \( 3 - K \) must be greater than 0, which means \( K < 3 \).

5. **Combining Conditions**:
From both cases, we find that for real solutions to exist:
\[ -3 < K < 3 \]

6. **Finding the Number of Solutions**:
- If \( K = 0 \):
\[ |e^{|x|} - 3| = 0 \implies e^{|x|} = 3 \]
This gives two solutions: \( |x| = \ln(3) \) or \( x = \pm \ln(3) \) (2 solutions).

- If \( K = 1 \):
\[ |e^{|x|} - 3| = 1 \implies e^{|x|} = 4 \quad \text{or} \quad e^{|x|} = 2 \]
This gives four solutions: \( |x| = \ln(4) \) (2 solutions) and \( |x| = \ln(2) \) (2 solutions).

- If \( K = 2 \):
\[ |e^{|x|} - 3| = 2 \implies e^{|x|} = 5 \quad \text{or} \quad e^{|x|} = 1 \]
This gives three solutions: \( |x| = \ln(5) \) (2 solutions) and \( |x| = 0 \) (1 solution).

- If \( K \) is in the range \( 0 < K < 2 \):
The graph will intersect the horizontal line \( y = K \) at two points, giving a total of 2 solutions.

- If \( K \) is in the range \( 2 < K < 3 \):
The graph will intersect the horizontal line \( y = K \) at one point, giving a total of 2 solutions.

### Summary of Solutions:
- For \( K = 0 \): 2 solutions
- For \( K = 1 \): 4 solutions
- For \( K = 2 \): 3 solutions
- For \( 0 < K < 2 \): 2 solutions
- For \( 2 < K < 3 \): 2 solutions

### Conclusion:
The number of solutions to \( |e^{|x|} - 3| = K \) depends on the value of \( K \) as described above.