Question

Which of the following functions is an odd function?

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

Answer 1

The Correct Option is D

To determine which function is odd, we need to understand that a function \( f(x) \) is odd if it satisfies the condition \( f(-x) = -f(x) \) for all \( x \) in its domain. Let's evaluate the given options:

Option (a): \( f(x) = 2^{-xx} \)

Let \( f(x) = 2^{-(x^2)} \), a quadratic function, which can be rewritten as:

\( f(-x) = 2^{-((-x)^2)} = 2^{-(x^2)} = f(x) \)

Since \( f(-x) = f(x) \), this is an even function, not odd.

Option (b): \( f(x) = 2^{x-xxxx} \)

Assuming \( f(x) = 2^{x-x^4} \), let's check:

\( f(-x) = 2^{-x-(-x)^4} = 2^{-x-x^4} \)

We find that \( f(-x) \neq -f(x) \), as \( 2^{-x-x^4} \neq -2^{x-x^4} \); thus, it isn't odd.

Conclusion:

Both functions fail the odd function condition. Therefore, the answer is that neither (a) nor (b) is an odd function.