If the graph of the function `f(x)=(a^(x)-1)/(x^(n)(a^(x)+1))` is symmetric about `y`-axis then `n` is equal to

Options

A. `1//5`
B. `1//3`
C. `1//4`
D. `-1//3`

Explanation

To determine the value of \( n \) such that the function

\[ f(x) = \frac{a^x - 1}{x^n (a^x + 1)} \]

is symmetric about the \( y \)-axis, we need to check the condition for even functions. A function is symmetric about the \( y \)-axis if \( f(-x) = f(x) \).

### Step-by-Step Solution:

1. **Find \( f(-x) \)**:
Substitute \(-x\) into the function:
\[ f(-x) = \frac{a^{-x} - 1}{(-x)^n (a^{-x} + 1)} \]

2. **Simplify \( f(-x) \)**:
We can rewrite \( a^{-x} \) as \( \frac{1}{a^x} \):
\[ f(-x) = \frac{\frac{1}{a^x} - 1}{(-x)^n \left(\frac{1}{a^x} + 1\right)} = \frac{\frac{1 - a^x}{a^x}}{(-x)^n \left(\frac{1 + a^x}{a^x}\right)} \]
Simplifying further:
\[ f(-x) = \frac{1 - a^x}{(-x)^n (1 + a^x)} \]

3. **Rearranging \( f(-x) \)**:
We can express \( (-x)^n \) as \( (-1)^n x^n \):
\[ f(-x) = \frac{1 - a^x}{(-1)^n x^n (1 + a^x)} \]

4. **Set \( f(-x) \) equal to \( f(x) \)**:
We need to equate \( f(-x) \) and \( f(x) \):
\[ \frac{1 - a^x}{(-1)^n x^n (1 + a^x)} = \frac{a^x - 1}{x^n (a^x + 1)} \]

5. **Cross-multiply to eliminate fractions**:
Cross-multiplying gives:
\[ (1 - a^x)(x^n (a^x + 1)) = (a^x - 1)(-1)^n x^n (1 + a^x) \]

6. **Simplify both sides**:
The left side becomes:
\[ x^n (1 - a^x)(a^x + 1) = x^n (a^x + 1 - a^{2x} - a^x) = x^n (1 - a^{2x}) \]
The right side simplifies to:
\[ (-1)^n x^n (a^x - 1)(1 + a^x) = (-1)^n x^n (a^{2x} - 1) \]

7. **Equate the simplified forms**:
Now we have:
\[ x^n (1 - a^{2x}) = (-1)^n x^n (a^{2x} - 1) \]

8. **Divide both sides by \( x^n \) (assuming \( x \neq 0 \))**:
\[ 1 - a^{2x} = (-1)^n (a^{2x} - 1) \]

9. **Rearranging gives**:
\[ 1 - a^{2x} = (-1)^n a^{2x} - (-1)^n \]
Rearranging this leads to:
\[ 1 + (-1)^n = a^{2x} (1 + (-1)^n) \]

10. **Analyze the equation**:
For this equation to hold for all \( x \), the term \( 1 + (-1)^n \) must equal zero, which implies that \( n \) must be odd.

### Conclusion:
Thus, the value of \( n \) must be an odd integer.