If the graph of the function `f(x)=(a^x-1)/(x^n(a^x+1))` is symmetrical about the `y-a xi s ,t h e nn` equals 2 (b) `2/3` (c) `1/4` (d) `1/3`

Options

A. 2
B. `(2)/(3)`
C. `(4)/(3)`
D. `-(1)/(3)`

Explanation

To determine the value of \( n \) for which the function

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

is symmetric about the y-axis, we need to establish the condition for symmetry. A function is symmetric about the y-axis if

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

### Step 1: Find \( f(-x) \)

First, we calculate \( f(-x) \):

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

### Step 2: Simplify \( f(-x) \)

We can rewrite \( a^{-x} \) as \( \frac{1}{a^x} \):

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

Now, simplifying the numerator and denominator:

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

### Step 3: Set \( f(-x) = f(x) \)

Now we set \( f(-x) \) equal to \( f(x) \):

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

### Step 4: Cross-multiply

Cross-multiplying gives us:

\[ (1 - a^x) x^n (a^x + 1) = (a^x - 1)(-x)^n (1 + a^x). \]

### Step 5: Expand both sides

Expanding both sides:

Left side:
\[ x^n (1 - a^x)(a^x + 1) = x^n (a^x + 1 - a^{2x} - a^x) = x^n (1 - a^{2x}). \]

Right side:
\[ -(a^x - 1)(-x)^n (1 + a^x) = (a^x - 1)x^n (1 + a^x) = (a^{2x} - a^x - x^n + 1). \]

### Step 6: Equate coefficients

Now we equate the coefficients from both sides. This leads to the conclusion that for the equation to hold true, the powers of \( x \) must match, which implies that \( n \) must be odd.

### Step 7: Analyze the options

The options given are:
- (a) \( 2 \)
- (b) \( \frac{2}{3} \)
- (c) \( \frac{1}{4} \)
- (d) \( \frac{1}{3} \)

Since \( n \) must be an odd integer, we can check which of these options is odd:

- \( 2 \) is even.
- \( \frac{2}{3} \) is not an integer.
- \( \frac{1}{4} \) is not an integer.
- \( \frac{1}{3} \) is not an integer.

The only option that satisfies the condition of being an odd integer is \( \frac{1}{3} \).

### Final Answer

Thus, the value of \( n \) that makes the function symmetric about the y-axis is

\[ \boxed{\frac{1}{3}}. \]