Which of the following functions is an odd function?

Options

A. `2^(-x.x)`
B. `2^(x-x.x.x.x)`
C. Both (a) and (b)
D. Neither (a) nor (b)

Answer

To determine which of the given functions is an odd function, we will follow these steps:

### Step 1: Understand the definitions of odd and even functions
- An **odd function** satisfies the condition: \( f(-x) = -f(x) \).
- An **even function** satisfies the condition: \( f(-x) = f(x) \).

### Step 2: Analyze the first function
Let’s denote the first function as \( f(x) = 2^{-x} \cdot x \).

1. Calculate \( f(-x) \):
\[ f(-x) = 2^{-(-x)} \cdot (-x) = 2^{x} \cdot (-x) = -2^{x} \cdot x \]

2. Check if it satisfies the odd function condition:
\[ -f(x) = - (2^{-x} \cdot x) = -2^{-x} \cdot x \]
Since \( f(-x) = -2^{x} \cdot x \) does not equal \( -2^{-x} \cdot x \), the first function is **not odd**.

### Step 3: Analyze the second function
Let’s denote the second function as \( g(x) = 2^{x} \cdot (-x) \cdot x \).

1. Calculate \( g(-x) \):
\[ g(-x) = 2^{-x} \cdot (-(-x)) \cdot (-x) = 2^{-x} \cdot x \cdot (-x) = -2^{-x} \cdot x^2 \]

2. Check if it satisfies the odd function condition:
\[ -g(x) = - (2^{x} \cdot (-x) \cdot x) = 2^{x} \cdot x^2 \]
Since \( g(-x) = -2^{-x} \cdot x^2 \) does not equal \( 2^{x} \cdot x^2 \), the second function is **not odd**.

### Step 4: Conclusion
Since neither of the functions satisfies the condition for being an odd function, we conclude that the answer is **neither a nor b**.

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