Which of the following function is an odd function?

Options

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

Answer

To determine which of the given functions is an odd function, we need to analyze each function based on the definition of odd functions. A function \( f(x) \) is considered odd if:

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

Let's evaluate each option step by step.

### Step 1: Evaluate \( f(x) = 2^{-x^2} \)

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

2. Compare \( f(-x) \) with \( -f(x) \):
\[ -f(x) = -2^{-x^2} \]
Since \( f(-x) = 2^{-x^2} \) is not equal to \( -2^{-x^2} \), this function is **not odd**.

### Step 2: Evaluate \( f(x) = -2^{-x^2} \)

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

2. Compare \( f(-x) \) with \( -f(x) \):
\[ -f(x) = -(-2^{-x^2}) = 2^{-x^2} \]
Since \( f(-x) = -2^{-x^2} \) is equal to \( -f(x) \), this function is **not odd**.

### Step 3: Evaluate \( f(x) = \cos x \)

1. Calculate \( f(-x) \):
\[ f(-x) = \cos(-x) = \cos x \]

2. Compare \( f(-x) \) with \( -f(x) \):
\[ -f(x) = -\cos x \]
Since \( f(-x) = \cos x \) is not equal to \( -\cos x \), this function is **not odd**.

### Conclusion

After evaluating all the options, we find that none of the functions provided are odd functions. Therefore, the answer is:

**None of the options are odd functions.**

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