The number of roots of the equation
`1+log_(2)(1-x)=2^(-x)`, is

Options

A. 0
B. 1
C. 2
D. may

Answer

To find the number of roots of the equation \(1 + \log_{2}(1 - x) = 2^{-x}\), we will analyze both sides of the equation graphically.

### Step-by-Step Solution:

1. **Rearranging the Equation**:
We start with the equation:
\[ 1 + \log_{2}(1 - x) = 2^{-x} \]
We can rewrite this as:
\[ \log_{2}(1 - x) = 2^{-x} - 1 \]

2. **Identifying the Functions**:
Let:
\[ f(x) = \log_{2}(1 - x) \]
and
\[ g(x) = 2^{-x} - 1 \]
We need to find the number of intersections between the graphs of \(f(x)\) and \(g(x)\).

3. **Domain of \(f(x)\)**:
The function \(f(x) = \log_{2}(1 - x)\) is defined for:
\[ 1 - x > 0 \implies x < 1 \]
Therefore, the domain of \(f(x)\) is \(x < 1\).

4. **Behavior of \(f(x)\)**:
- As \(x\) approaches \(1\) from the left, \(f(x) \to -\infty\).
- At \(x = 0\), \(f(0) = \log_{2}(1) = 0\).

5. **Behavior of \(g(x)\)**:
The function \(g(x) = 2^{-x} - 1\):
- As \(x \to -\infty\), \(g(x) \to \infty\).
- At \(x = 0\), \(g(0) = 2^{0} - 1 = 0\).
- As \(x \to \infty\), \(g(x) \to -1\).

6. **Graphing the Functions**:
- The graph of \(f(x)\) is a decreasing function starting from \(0\) at \(x = 0\) and going to \(-\infty\) as \(x\) approaches \(1\).
- The graph of \(g(x)\) starts from \(0\) at \(x = 0\) and approaches \(-1\) as \(x\) increases.

7. **Finding Intersections**:
- Both functions intersect at \(x = 0\).
- Since \(f(x)\) is decreasing and \(g(x)\) is decreasing but starts from a higher value, they will intersect again at some point before \(g(x)\) goes below \(f(x)\).

8. **Conclusion**:
By analyzing the graphs and their behaviors, we can conclude that there are **two points of intersection** between \(f(x)\) and \(g(x)\).

### Final Answer:
The number of roots of the equation \(1 + \log_{2}(1 - x) = 2^{-x}\) is **2**.