Let `f(x)=2^(-x)` and `g(x)=4. 2^(-x)` Which is true ?

Options

A. g(x)=f(x-2)
B. g(x)=f(x+2)
C. g(x)=f(x)-2
D. g(x)=f(x)+2

Answer

To solve the problem, we need to analyze the functions given:

1. **Define the functions**:
- \( f(x) = 2^{-x} \)
- \( g(x) = 4 \cdot 2^{-x} \)

2. **Rewrite \( g(x) \)**:
- We can express \( g(x) \) in terms of powers of 2:
\[ g(x) = 4 \cdot 2^{-x} = 2^2 \cdot 2^{-x} = 2^{2 - x} \]

3. **Relate \( g(x) \) to \( f(x) \)**:
- We know that \( f(x) = 2^{-x} \).
- To express \( g(x) \) in terms of \( f(x) \), we can rewrite \( g(x) \):
\[ g(x) = 2^{2 - x} = 2^{-(x - 2)} = 2^{-x + 2} \]
- This can also be expressed as:
\[ g(x) = 2^{-x} \cdot 2^2 = 4 \cdot f(x) \]

4. **Finding the relationship**:
- We want to find if \( g(x) \) can be expressed as \( f(x) \) plus or minus some constant.
- We can express \( g(x) \) in terms of \( f(x) \):
\[ g(x) = 4 \cdot f(x) \]
- This does not directly match any of the options given, so we need to check the options.

5. **Check the options**:
- Option A: \( g(x) = f(x) - 2 \)
- Option B: \( g(x) = f(x) + 2 \)
- Option C: \( g(x) = f(x) - 2 \) (repeated)
- Option D: \( g(x) = f(x) + 2 \) (repeated)

6. **Conclusion**:
- Since \( g(x) = 4 \cdot f(x) \), we can conclude that none of the options directly relate \( g(x) \) to \( f(x) \) with a simple addition or subtraction of 2.
- However, if we take \( g(x) = f(x) - 2 \), we can see that this is not true. The correct relationship is \( g(x) = 4 \cdot f(x) \).

Thus, the correct answer is not among the provided options, but the closest relationship we found is \( g(x) = 4 \cdot f(x) \).