The function f(x) is defined over the whole number scale by the following law:
`f(x)={{:(,2x^(3)+1, if x le 2),(,1//(x-2), if 2 lt x le 3),(,2x-5, if x gt 3):}`
Find `f(sqrt2), f(sqrt8), f(sqrt(log)2) ,f(1024)`.

Answer

To solve the problem, we need to evaluate the function \( f(x) \) for the given values \( \sqrt{2} \), \( \sqrt{8} \), \( \sqrt{\log 2} \), and \( 1024 \). The function \( f(x) \) is defined piecewise as follows:

\[ f(x) = \begin{cases} 2x^3 + 1 & \text{if } x \leq 2 \\ \frac{1}{x - 2} & \text{if } 2 < x \leq 3 \\ 2x - 5 & \text{if } x > 3 \end{cases} \]

### Step 1: Evaluate \( f(\sqrt{2}) \)

1. Calculate \( \sqrt{2} \):
\[ \sqrt{2} \approx 1.414 \]
Since \( \sqrt{2} \leq 2 \), we use the first case of the function:
\[ f(\sqrt{2}) = 2(\sqrt{2})^3 + 1 \]
2. Calculate \( (\sqrt{2})^3 \):
\[ (\sqrt{2})^3 = 2\sqrt{2} \]
3. Substitute back into the function:
\[ f(\sqrt{2}) = 2(2\sqrt{2}) + 1 = 4\sqrt{2} + 1 \]

### Step 2: Evaluate \( f(\sqrt{8}) \)

1. Calculate \( \sqrt{8} \):
\[ \sqrt{8} = 2\sqrt{2} \approx 2.828 \]
Since \( 2 < \sqrt{8} \leq 3 \), we use the second case of the function:
\[ f(\sqrt{8}) = \frac{1}{\sqrt{8} - 2} \]
2. Simplify \( \sqrt{8} - 2 \):
\[ \sqrt{8} - 2 = 2\sqrt{2} - 2 = 2(\sqrt{2} - 1) \]
3. Substitute back into the function:
\[ f(\sqrt{8}) = \frac{1}{2(\sqrt{2} - 1)} = \frac{1}{2\sqrt{2} - 2} \]

### Step 3: Evaluate \( f(\sqrt{\log 2}) \)

1. Calculate \( \sqrt{\log 2} \):
\[ \log 2 \approx 0.301 \quad \Rightarrow \quad \sqrt{\log 2} \approx 0.55 \]
Since \( \sqrt{\log 2} \leq 2 \), we use the first case of the function:
\[ f(\sqrt{\log 2}) = 2(\sqrt{\log 2})^3 + 1 \]
2. Calculate \( (\sqrt{\log 2})^3 \):
\[ (\sqrt{\log 2})^3 = (\log 2)^{3/2} \]
3. Substitute back into the function:
\[ f(\sqrt{\log 2}) = 2(\log 2)^{3/2} + 1 \]

### Step 4: Evaluate \( f(1024) \)

1. Since \( 1024 > 3 \), we use the third case of the function:
\[ f(1024) = 2(1024) - 5 \]
2. Calculate:
\[ f(1024) = 2048 - 5 = 2043 \]

### Final Results

- \( f(\sqrt{2}) = 4\sqrt{2} + 1 \)
- \( f(\sqrt{8}) = \frac{1}{2\sqrt{2} - 2} \)
- \( f(\sqrt{\log 2}) = 2(\log 2)^{3/2} + 1 \)
- \( f(1024) = 2043 \)