A functional equation is an equation, which relates the values assumed by a function at two or more points, which are themselves related in a particular manner. For example, we define an odd function by the relation `f(-x) = -f(x)` for all x.This defination can be paraphrased to say that it is a function f(x), which satisfies the functional relation f(x) + f(y) = 0, whenever x+y = 0. Of course this does not identify the function uniquely, sometimes with some additionl information, a function satisfying a given functional equation can be identified uniquely.
Suppose a functional equation has a relation between f(x) and `f(1/x)`, then due to the reason that reciprocal of a reciprocal gives back the original number, we can substitute `1/x` for x. This will result into another equation and solving these two, we can find f(x) uniquely. Similarly, we can solve an equation which contains f(x) and f(-x). Such equations are of repetitive nature .
In the functional equation `af(x) + bf(-x) = g(x)`, if a + b = 0, then f(x) is equal to

Options

A. 0
B. `(x-1/x)/(2a)`
C. `(x+1/x)/(2a)`
D. not possible

Answer

To solve the functional equation given by \( af(x) + bf(-x) = g(x) \) under the condition that \( a + b = 0 \), we can follow these steps:

### Step 1: Understand the Given Condition
We are given that \( a + b = 0 \). This implies that \( b = -a \).

### Step 2: Substitute \( b \) in the Functional Equation
Substituting \( b = -a \) into the functional equation gives:
\[ af(x) - af(-x) = g(x) \]

### Step 3: Factor Out \( a \)
We can factor out \( a \) from the left-hand side:
\[ a(f(x) - f(-x)) = g(x) \]

### Step 4: Analyze the Equation
From the equation \( a(f(x) - f(-x)) = g(x) \), we can see that if \( a \neq 0 \), we can divide both sides by \( a \):
\[ f(x) - f(-x) = \frac{g(x)}{a} \]

### Step 5: Consider the Case When \( a = 0 \)
If \( a = 0 \), then from the original equation \( af(x) + bf(-x) = g(x) \) simplifies to:
\[ bf(-x) = g(x) \]
Since \( b = -a \) and \( a = 0 \), this means \( b \) can be any value, and we cannot determine \( f(x) \) uniquely.

### Step 6: Conclusion
Thus, if \( a + b = 0 \) and \( a \neq 0 \), we can express \( f(x) \) in terms of \( g(x) \):
\[ f(x) = f(-x) + \frac{g(x)}{a} \]
However, if \( a = 0 \), \( f(x) \) cannot be determined uniquely.

### Final Answer
Hence, the solution for \( f(x) \) is:
\[ f(x) = f(-x) + \frac{g(x)}{a} \quad \text{(for } a \neq 0\text{)} \]