Examine whether the function are even or odd or none.
`f(x) = ((1 + 2^(x))^(7))/(2^(x))`

Answer

To determine whether the function \( f(x) = \frac{(1 + 2^x)^7}{2^x} \) is even, odd, or neither, we will follow these steps:

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

We start by substituting \(-x\) into the function:

\[ f(-x) = \frac{(1 + 2^{-x})^7}{2^{-x}} \]

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

Next, we simplify \( f(-x) \):

\[ f(-x) = \frac{(1 + \frac{1}{2^x})^7}{\frac{1}{2^x}} = (1 + \frac{1}{2^x})^7 \cdot 2^x \]

### Step 3: Further simplify \( f(-x) \)

Now, let's simplify \( (1 + \frac{1}{2^x})^7 \):

\[ f(-x) = (1 + \frac{1}{2^x})^7 \cdot 2^x = \left( \frac{2^x + 1}{2^x} \right)^7 \cdot 2^x = \frac{(2^x + 1)^7}{(2^x)^7} \cdot 2^x \]

This simplifies to:

\[ f(-x) = \frac{(2^x + 1)^7}{2^{7x}} \cdot 2^x = \frac{(2^x + 1)^7}{2^{6x}} \]

### Step 4: Compare \( f(-x) \) with \( f(x) \)

Now we will compare \( f(-x) \) with \( f(x) \):

1. We have \( f(x) = \frac{(1 + 2^x)^7}{2^x} \)
2. We have \( f(-x) = \frac{(2^x + 1)^7}{2^{6x}} \)

Notice that \( (1 + 2^x)^7 = (2^x + 1)^7 \), so we can say:

\[ f(-x) = \frac{(1 + 2^x)^7}{2^{6x}} \]

### Step 5: Determine if the function is even, odd, or neither

Now we check the conditions for even and odd functions:

- A function is **even** if \( f(-x) = f(x) \).
- A function is **odd** if \( f(-x) = -f(x) \).

From our calculations:

1. \( f(-x) = \frac{(1 + 2^x)^7}{2^{6x}} \)
2. \( f(x) = \frac{(1 + 2^x)^7}{2^x} \)

Clearly, \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \).

### Conclusion

Since neither condition for even or odd functions is satisfied, we conclude that the function \( f(x) \) is **neither even nor odd**.

---