If `f^2(x)*f((1-x)/(1+x))=x^3, [x!=-1,1 and f(x)!=0],` then find `|[f(-2)]|` (where [] is the greatest integer function).

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the given equation
We start with the equation:
\[ f^2(x) \cdot f\left(\frac{1-x}{1+x}\right) = x^3 \]
where \( x \neq -1, 1 \) and \( f(x) \neq 0 \).

### Step 2: Substitute \( x \) with \( \frac{1-x}{1+x} \)
Next, we replace \( x \) in the original equation with \( \frac{1-x}{1+x} \):
\[ f^2\left(\frac{1-x}{1+x}\right) \cdot f\left(\frac{1 - \frac{1-x}{1+x}}{1 + \frac{1-x}{1+x}}\right) = \left(\frac{1-x}{1+x}\right)^3 \]

### Step 3: Simplify the substitution
Now we simplify the second term:
\[ \frac{1 - \frac{1-x}{1+x}}{1 + \frac{1-x}{1+x}} = \frac{\frac{(1+x) - (1-x)}{1+x}}{\frac{(1+x) + (1-x)}{1+x}} = \frac{\frac{2x}{1+x}}{\frac{2}{1+x}} = x \]
Thus, the equation becomes:
\[ f^2\left(\frac{1-x}{1+x}\right) \cdot f(x) = \left(\frac{1-x}{1+x}\right)^3 \]

### Step 4: Define the new equation
Let’s denote this new equation as (2):
\[ f^2\left(\frac{1-x}{1+x}\right) \cdot f(x) = \left(\frac{1-x}{1+x}\right)^3 \]

### Step 5: Divide the two equations
Now we have two equations:
1. \( f^2(x) \cdot f\left(\frac{1-x}{1+x}\right) = x^3 \)
2. \( f^2\left(\frac{1-x}{1+x}\right) \cdot f(x) = \left(\frac{1-x}{1+x}\right)^3 \)

Dividing equation (1) by equation (2):
\[ \frac{f^2(x) \cdot f\left(\frac{1-x}{1+x}\right)}{f^2\left(\frac{1-x}{1+x}\right) \cdot f(x)} = \frac{x^3}{\left(\frac{1-x}{1+x}\right)^3} \]
This simplifies to:
\[ \frac{f(x)}{f\left(\frac{1-x}{1+x}\right)} = \frac{x^3}{\left(\frac{1-x}{1+x}\right)^3} \]

### Step 6: Solve for \( f(x) \)
From the above equation, we can deduce:
\[ f(x) = k \cdot x^3 \]
where \( k \) is a constant.

### Step 7: Find \( f(-2) \)
Now, we need to find \( f(-2) \):
\[ f(-2) = k \cdot (-2)^3 = k \cdot (-8) \]

### Step 8: Determine the value of \( k \)
Since the function must satisfy the original equation, we can choose \( k = 1 \) for simplicity:
\[ f(-2) = -8 \]

### Step 9: Find the greatest integer function and absolute value
Now, we need to find \( |[f(-2)]| \):
\[ [f(-2)] = [-8] = -8 \]
Thus,
\[ |[f(-2)]| = | -8 | = 8 \]

### Final Answer
The final answer is:
\[ \boxed{8} \]