If f(x) = `e^(-x)`, then `(f (-a))/(f(b))` is equal to

Options

A. f(a + b)
B. f (a - b)
C. f (-a + b)
D. f(-a -b)

Answer

To solve the problem, we need to find the value of \(\frac{f(-a)}{f(b)}\) given that \(f(x) = e^{-x}\).

### Step-by-Step Solution:

1. **Find \(f(-a)\)**:
\[ f(-a) = e^{-(-a)} = e^{a} \]

2. **Find \(f(b)\)**:
\[ f(b) = e^{-b} \]

3. **Set up the fraction**:
\[ \frac{f(-a)}{f(b)} = \frac{e^{a}}{e^{-b}} \]

4. **Simplify the fraction**:
Using the property of exponents \(\frac{e^m}{e^n} = e^{m-n}\):
\[ \frac{e^{a}}{e^{-b}} = e^{a - (-b)} = e^{a + b} \]

5. **Express in terms of \(f\)**:
We know that \(f(x) = e^{-x}\), so:
\[ f(a + b) = e^{-(a + b)} = e^{a + b} \]

Thus, we can conclude that:
\[ \frac{f(-a)}{f(b)} = e^{a + b} = f(-a - b) \]

### Final Answer:
The expression \(\frac{f(-a)}{f(b)}\) is equal to \(f(a + b)\).