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

### Step 3: Simplify Each Component
1. **Sine Function**:
\[ \sin((-x)^7) = \sin(-x^7) = -\sin(x^7) \quad \text{(since sine is an odd function)} \]
2. **Exponential Function**:
\[ e^{(-x)^5} = e^{-x^5} \quad \text{(exponential function is even)} \]
3. **Sign Function**:
\[ \text{sgn}((-x)^9) = \text{sgn}(-x^9) = -\text{sgn}(x^9) \quad \text{(sign function is odd)} \]

### Step 4: Combine the Components
Now, substituting these back into \( f(-x) \):
\[ f(-x) = (-\sin(x^7)) \cdot e^{-x^5} \cdot (-\text{sgn}(x^9)) \]
This simplifies to:
\[ f(-x) = \sin(x^7) \cdot e^{-x^5} \cdot \text{sgn}(x^9) \]

### Step 5: Compare \( f(-x) \) with \( f(x) \)
Now we compare \( f(-x) \) with \( f(x) \):
\[ f(x) = \sin(x^7) \cdot e^{x^5} \cdot \text{sgn}(x^9) \]
We can see that:
\[ f(-x) = \sin(x^7) \cdot e^{-x^5} \cdot \text{sgn}(x^9) \]

### Step 6: Determine the Relationship
From the comparison:
\[ f(-x) \neq f(x) \quad \text{(not even)} \]
\[ f(-x) = -\left(\sin(x^7) \cdot e^{x^5} \cdot \text{sgn}(x^9)\right) = -f(x) \quad \text{(odd)} \]

### Conclusion
Since \( f(-x) = -f(x) \), we conclude that the function \( f(x) \) is an **odd function**.

### Final Answer
The function \( f(x) \) is an **odd function**.

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