If f is an even function, then find the realvalues of x satisfying the equation `f(x)=f((x+1)/(x+2))`

Answer

To solve the equation \( f(x) = f\left(\frac{x+1}{x+2}\right) \) given that \( f \) is an even function, we can follow these steps:

### Step 1: Use the property of even functions
Since \( f \) is an even function, we know that:
\[ f(x) = f(-x) \]
This means that \( f(x) = f\left(\frac{x+1}{x+2}\right) \) can also be expressed as:
\[ f\left(\frac{x+1}{x+2}\right) = f\left(-\frac{x+1}{x+2}\right) \]

### Step 2: Set up the equation
From the property of even functions, we can set up the following equation:
\[ x = -\frac{x+1}{x+2} \]

### Step 3: Clear the fraction
To eliminate the fraction, we can multiply both sides by \( x + 2 \):
\[ x(x + 2) = - (x + 1) \]
This simplifies to:
\[ x^2 + 2x = -x - 1 \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ x^2 + 2x + x + 1 = 0 \]
which simplifies to:
\[ x^2 + 3x + 1 = 0 \]

### Step 5: Solve the quadratic equation
Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = 3, c = 1 \):
\[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{-3 \pm \sqrt{9 - 4}}{2} \]
\[ x = \frac{-3 \pm \sqrt{5}}{2} \]

### Step 6: Find the solutions
Thus, the solutions for \( x \) are:
\[ x = \frac{-3 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{5}}{2} \]

### Final Answer
The real values of \( x \) satisfying the equation \( f(x) = f\left(\frac{x+1}{x+2}\right) \) are:
\[ x = \frac{-3 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{5}}{2} \]