Classify the following functions for being even or odd:
` ( a^x - a^(-x))/( a^x +a^(-x))`

Answer

To classify the function \( f(x) = \frac{a^x - a^{-x}}{a^x + a^{-x}} \) as even or odd, we will follow these steps:

### Step 1: Define the function
Let \( f(x) = \frac{a^x - a^{-x}}{a^x + a^{-x}} \).

### Step 2: Find \( f(-x) \)
To determine if the function is even or odd, we first need to calculate \( f(-x) \):
\[ f(-x) = \frac{a^{-x} - a^{x}}{a^{-x} + a^{x}} \]

### Step 3: Simplify \( f(-x) \)
We can rewrite \( a^{-x} \) as \( \frac{1}{a^x} \):
\[ f(-x) = \frac{\frac{1}{a^x} - a^{x}}{\frac{1}{a^x} + a^{x}} \]
Now, multiply the numerator and the denominator by \( a^x \) to eliminate the fractions:
\[ f(-x) = \frac{1 - a^{2x}}{1 + a^{2x}} \]

### Step 4: Compare \( f(-x) \) with \( -f(x) \)
Next, we need to find \( -f(x) \):
\[ -f(x) = -\left(\frac{a^x - a^{-x}}{a^x + a^{-x}}\right) = \frac{-(a^x - a^{-x})}{a^x + a^{-x}} = \frac{a^{-x} - a^{x}}{a^x + a^{-x}} \]
Notice that this is the same as \( f(-x) \):
\[ f(-x) = -f(x) \]

### Step 5: Conclusion
Since \( f(-x) = -f(x) \), we conclude that the function \( f(x) \) is an **odd function**.

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