Examine whether the following function are even or odd or neither even nor odd, where [ ] denotes greatest integer function.
`f(x)=((1+2^(x))^(7))/(2^(x))`

Explanation

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: Define the function
The given function is:
\[ f(x) = \frac{(1 + 2^x)^7}{2^x} \]

### Step 2: Find \( f(-x) \)
To check if the function is even or odd, we need to calculate \( f(-x) \):
\[ f(-x) = \frac{(1 + 2^{-x})^7}{2^{-x}} \]

### Step 3: Simplify \( f(-x) \)
We can simplify \( f(-x) \):
\[ f(-x) = \frac{(1 + \frac{1}{2^x})^7}{\frac{1}{2^x}} = (1 + \frac{1}{2^x})^7 \cdot 2^x \]
Next, we can rewrite \( (1 + \frac{1}{2^x})^7 \):
\[ = (1 + 2^{-x})^7 = \left(\frac{2^x + 1}{2^x}\right)^7 = \frac{(2^x + 1)^7}{(2^x)^7} \]
Thus, we have:
\[ f(-x) = \frac{(2^x + 1)^7}{(2^x)^7} \cdot 2^x = \frac{(2^x + 1)^7}{2^{7x}} \cdot 2^x = \frac{(2^x + 1)^7}{2^{6x}} \]

### Step 4: Compare \( f(x) \) and \( f(-x) \)
Now we compare \( f(x) \) and \( f(-x) \):
- \( f(x) = \frac{(1 + 2^x)^7}{2^x} \)
- \( f(-x) = \frac{(2^x + 1)^7}{2^{6x}} \)

We can see that:
\[ f(-x) \neq f(x) \]
since the denominators are different and the numerators are equivalent but not in the same form.

### Step 5: Check if \( f(-x) = -f(x) \)
Next, we check if \( f(-x) = -f(x) \):
\[ f(-x) = \frac{(2^x + 1)^7}{2^{6x}} \quad \text{and} \quad -f(x) = -\frac{(1 + 2^x)^7}{2^x} \]
Clearly, \( f(-x) \) is not equal to \( -f(x) \).

### Conclusion
Since \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \), we conclude that the function \( f(x) \) is neither even nor odd.

### Final Answer
The function \( f(x) \) is neither even nor odd.

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