After a single sheet of paper is folded in half, there are two layers of paper. The same sheet of paper is repeatly folded in half. If function f represents the number of layers of paper that result when the original sheet of paper is folded a total of x times, then which equation could represents this function?

Options

A. `f(x)=2x`
B. `f(x)=x^(2)`
C. `f(x)=2^(x)`
D. `f(x)=4^((x)/(2))`

Answer

To solve the problem of determining the function \( f \) that represents the number of layers of paper after folding a single sheet of paper \( x \) times, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Folding Process**:
- When you fold a single sheet of paper in half for the first time, you create 2 layers.
- Each subsequent fold doubles the number of layers from the previous fold.

2. **Identifying the Pattern**:
- After 1 fold:
\[ f(1) = 2 \text{ layers} \]
- After 2 folds:
\[ f(2) = 2 \times 2 = 4 \text{ layers} \]
- After 3 folds:
\[ f(3) = 2 \times 4 = 8 \text{ layers} \]
- After 4 folds:
\[ f(4) = 2 \times 8 = 16 \text{ layers} \]

3. **Generalizing the Function**:
- From the pattern observed, we can see that the number of layers after \( x \) folds is \( 2^x \).
- Therefore, we can express the function \( f \) as:
\[ f(x) = 2^x \]

4. **Verifying the Function**:
- For \( x = 1 \):
\[ f(1) = 2^1 = 2 \]
- For \( x = 2 \):
\[ f(2) = 2^2 = 4 \]
- For \( x = 3 \):
\[ f(3) = 2^3 = 8 \]
- For \( x = 4 \):
\[ f(4) = 2^4 = 16 \]
- The values match our earlier calculations.

5. **Conclusion**:
- The function that represents the number of layers of paper after folding \( x \) times is:
\[ f(x) = 2^x \]