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If f (x) is an odd periodic function wit...

If f (x) is an odd periodic function with period 2, then f (4) equals

A

0

B

2

C

4

D

`-4`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to use the properties of the function \( f(x) \) given in the question. 1. **Understanding the properties of the function**: - \( f(x) \) is an **odd function**, which means that: \[ f(-x) = -f(x) \] - \( f(x) \) is also a **periodic function** with a period of 2, which means that: \[ f(x + 2) = f(x) \] 2. **Finding \( f(4) \)**: - Since \( f(x) \) has a period of 2, we can reduce \( 4 \) modulo \( 2 \): \[ f(4) = f(4 - 2) = f(2) \] - Now, we can reduce \( f(2) \) further: \[ f(2) = f(2 - 2) = f(0) \] 3. **Finding \( f(0) \)**: - Since \( f(x) \) is an odd function, we can substitute \( x = 0 \) into the odd function property: \[ f(-0) = -f(0) \implies f(0) = -f(0) \] - This implies: \[ 2f(0) = 0 \implies f(0) = 0 \] 4. **Conclusion**: - Therefore, we have: \[ f(4) = f(2) = f(0) = 0 \] - Thus, the final answer is: \[ f(4) = 0 \]
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