`f(x)=(sin x^(7)) e^(x^(5)). Sgn( x^(9))` is:

Options

A. an even function
B. an odd function
C. neither even nor odd
D. None of these

Answer

To determine whether the function \( f(x) = (\sin(x^7)) e^{(x^5)} \cdot \text{sgn}(x^9) \) is even, odd, or neither, we will follow these steps:

### Step 1: Understand the definitions
- A function \( f(x) \) is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function \( f(x) \) is **odd** if \( f(-x) = -f(x) \) for all \( x \).

### Step 2: Compute \( f(-x) \)
We need to find \( f(-x) \):
\[ f(-x) = \sin((-x)^7) e^{(-x)^5} \cdot \text{sgn}((-x)^9) \]

### Step 3: Simplify \( f(-x) \)
Now, simplify each part:
- \( \sin((-x)^7) = \sin(-x^7) = -\sin(x^7) \) (since sine is an odd function)
- \( e^{(-x)^5} = e^{-x^5} \) (this is an even function)
- \( \text{sgn}((-x)^9) = \text{sgn}(-x^9) = -\text{sgn}(x^9) \) (since the signum function is odd)

Putting it all together:
\[ f(-x) = (-\sin(x^7)) e^{-x^5} \cdot (-\text{sgn}(x^9)) \]
This simplifies to:
\[ f(-x) = \sin(x^7) e^{-x^5} \cdot \text{sgn}(x^9) \]

### Step 4: Compare \( f(-x) \) with \( f(x) \)
Now, compare \( f(-x) \) with \( f(x) \):
\[ f(x) = \sin(x^7) e^{x^5} \cdot \text{sgn}(x^9) \]
\[ f(-x) = \sin(x^7) e^{-x^5} \cdot \text{sgn}(x^9) \]

### Step 5: Analyze the results
Since \( f(-x) \) is not equal to \( f(x) \) and also not equal to \( -f(x) \) (because of the \( e^{-x^5} \) term), we conclude that the function is neither even nor odd.

### Final Answer
Thus, the function \( f(x) \) is **neither even nor odd**.

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