Determine whether the following functions are even or odd or neither even nor odd:
`f(x)=x((a^(x)-1)/(a^(x)+1))`

Explanation

To determine whether the function \( f(x) = x \left( \frac{a^x - 1}{a^x + 1} \right) \) is even, odd, or neither, we will follow these steps:

### Step 1: Define Even and Odd Functions
A function \( f(x) \) is:
- **Even** if \( f(-x) = f(x) \)
- **Odd** if \( f(-x) = -f(x) \)

### Step 2: Calculate \( f(-x) \)
We start by substituting \(-x\) into the function:
\[ f(-x) = -x \left( \frac{a^{-x} - 1}{a^{-x} + 1} \right) \]

### Step 3: Simplify \( f(-x) \)
We can rewrite \( a^{-x} \) as \( \frac{1}{a^x} \):
\[ f(-x) = -x \left( \frac{\frac{1}{a^x} - 1}{\frac{1}{a^x} + 1} \right) \]

Now, we can simplify the fraction:
\[ f(-x) = -x \left( \frac{1 - a^x}{1 + a^x} \right) \]

### Step 4: Factor Out the Negative Sign
We can factor out the negative sign:
\[ f(-x) = x \left( \frac{a^x - 1}{a^x + 1} \right) \]

### Step 5: Compare \( f(-x) \) with \( f(x) \)
Now we compare \( f(-x) \) with \( f(x) \):
\[ f(x) = x \left( \frac{a^x - 1}{a^x + 1} \right) \]
\[ f(-x) = x \left( \frac{a^x - 1}{a^x + 1} \right) \]

Since \( f(-x) = f(x) \), we conclude that the function is **even**.

### Conclusion
The function \( f(x) = x \left( \frac{a^x - 1}{a^x + 1} \right) \) is an **even function**.

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