Which of the following function is an odd function?

Options

A. `f(x)=2^(-x^(2))`
B. `f(x)=2^(x-x^(4))`.
C. `f(x)=cosx`
D. none of these

Answer

To determine which of the given functions is an odd function, we need to apply the definition of an odd function. A function \( f(x) \) is considered odd if it satisfies the condition:

\[ f(-x) = -f(x) \]

Now, let's analyze each option step by step.

### Step 1: Analyze Option 1
**Function:** \( f(x) = 2k^{-x^2} \)

**Find \( f(-x) \):**
\[ f(-x) = 2k^{-(-x)^2} = 2k^{-x^2} \]

**Compare \( f(-x) \) with \( -f(x) \):**
\[ -f(x) = -2k^{-x^2} \]

Since \( f(-x) = 2k^{-x^2} \) is not equal to \( -f(x) = -2k^{-x^2} \), this function is **not an odd function**.

### Step 2: Analyze Option 2
**Function:** \( f(x) = 2^x - x - x^4 \)

**Find \( f(-x) \):**
\[ f(-x) = 2^{-x} - (-x) - (-x)^4 = 2^{-x} + x - x^4 \]

**Compare \( f(-x) \) with \( -f(x) \):**
\[ -f(x) = - (2^x - x - x^4) = -2^x + x + x^4 \]

Since \( f(-x) = 2^{-x} + x - x^4 \) is not equal to \( -f(x) = -2^x + x + x^4 \), this function is **not an odd function**.

### Step 3: Analyze Option 3
**Function:** \( f(x) = \cos x \)

**Find \( f(-x) \):**
\[ f(-x) = \cos(-x) \]

Using the property of cosine:
\[ \cos(-x) = \cos x \]

**Compare \( f(-x) \) with \( -f(x) \):**
\[ -f(x) = -\cos x \]

Since \( f(-x) = \cos x \) is not equal to \( -f(x) = -\cos x \), this function is **not an odd function**.

### Conclusion
After analyzing all three options, none of the functions satisfy the condition for being an odd function. Therefore, the answer is that **none of the options provided is an odd function**.

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