Given `f(x)` is symmetrical about the line `x=1`, then find out the value of 'x' satisfying `f(x-1)=f((x)/(x+1))`

Answer

To solve the problem step by step, we need to find the value of \( x \) that satisfies the equation \( f(x-1) = f\left(\frac{x}{x+1}\right) \), given that the function \( f(x) \) is symmetrical about the line \( x = 1 \).

### Step 1: Understand the symmetry condition
Since \( f(x) \) is symmetrical about the line \( x = 1 \), we have the property:
\[ f(1 + a) = f(1 - a) \]
for any \( a \). This means that if we can express \( x-1 \) and \( \frac{x}{x+1} \) in terms of \( 1 + a \) and \( 1 - a \), we can use this property.

### Step 2: Set up the equation
We want to solve:
\[ f(x-1) = f\left(\frac{x}{x+1}\right) \]
Using the symmetry property, we can write:
\[ x - 1 = 1 - a \quad \text{and} \quad \frac{x}{x+1} = 1 + a \]
This implies:
\[ x - 1 = 1 - a \implies x = 2 - a \]
\[ \frac{x}{x+1} = 1 + a \implies \frac{2-a}{3-a} = 1 + a \]

### Step 3: Cross-multiply and simplify
Cross-multiplying gives us:
\[ (2 - a)(3 - a) = (1 + a)(3 - a) \]
Expanding both sides:
\[ 6 - 5a + a^2 = 3 - a + 3a - a^2 \]
This simplifies to:
\[ 6 - 5a + a^2 = 3 + 2a - a^2 \]

### Step 4: Rearranging the equation
Bringing all terms to one side:
\[ 6 - 3 - 5a - 2a + a^2 + a^2 = 0 \]
This simplifies to:
\[ 2a^2 - 7a + 3 = 0 \]

### Step 5: Solve the quadratic equation
Now we can use the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ a = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 2 \cdot 3}}{2 \cdot 2} \]
Calculating the discriminant:
\[ 49 - 24 = 25 \]
Thus, we have:
\[ a = \frac{7 \pm 5}{4} \]
Calculating the two possible values for \( a \):
1. \( a = \frac{12}{4} = 3 \)
2. \( a = \frac{2}{4} = \frac{1}{2} \)

### Step 6: Find corresponding \( x \) values
Using \( x = 2 - a \):
1. For \( a = 3 \): \( x = 2 - 3 = -1 \)
2. For \( a = \frac{1}{2} \): \( x = 2 - \frac{1}{2} = \frac{3}{2} \)

### Step 7: Check for additional values
We also consider \( a = 0 \) since \( f(x) \) is symmetrical:
- If \( a = 0 \), then \( x = 2 \).

### Final Values
Thus, the values of \( x \) that satisfy the equation are:
1. \( x = -1 \)
2. \( x = \frac{3}{2} \)
3. \( x = 2 \)

### Summary of Solution
The values of \( x \) satisfying \( f(x-1) = f\left(\frac{x}{x+1}\right) \) are \( -1, \frac{3}{2}, \) and \( 2 \).