Let `f(x+y)=f(y)andf(1)=2" for all "x,yinR` where f(x) is continuous function. What is f(1) equal to ?

Options

A. `2log_(e)2`
B. `log_(e)2`
C. 1
D. 0

Explanation

To solve the problem, we start with the functional equation given in the question:

**Step 1: Analyze the functional equation.**
We have the equation:
\[ f(x+y) = f(y) \]
for all \( x, y \in \mathbb{R} \).

**Hint:** This equation suggests that \( f \) is independent of \( x \). Try substituting specific values for \( x \) and \( y \) to explore the implications.

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**Step 2: Substitute \( y = 0 \) into the equation.**
Let \( y = 0 \):
\[ f(x + 0) = f(0) \]
This simplifies to:
\[ f(x) = f(0) \]
for all \( x \in \mathbb{R} \).

**Hint:** This indicates that \( f(x) \) is a constant function. What does that imply about the value of \( f(0) \)?

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**Step 3: Determine the value of the constant function.**
Since \( f(x) \) is constant for all \( x \), we can denote this constant as \( c \). Thus, we have:
\[ f(x) = c \]
for all \( x \in \mathbb{R} \).

**Hint:** Use the information given in the problem about \( f(1) \) to find the value of \( c \).

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**Step 4: Use the initial condition.**
From the problem, we know:
\[ f(1) = 2 \]
Since \( f(x) = c \) for all \( x \), we have:
\[ c = 2 \]

**Hint:** Now that you have found the constant, what is the final value of \( f(1) \)?

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**Step 5: Conclusion.**
Thus, we conclude:
\[ f(1) = 2 \]

**Final Answer:** \( f(1) = 2 \)

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