The function `f :R -> R` satisfies the condition `mf(x - 1) + nf(-x) = 2| x | + 1`. If `f(-2) = 5` and `f(1) = 1` find `m` and `n`

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) given the function \( f : \mathbb{R} \to \mathbb{R} \) that satisfies the equation:

\[ m f(x - 1) + n f(-x) = 2 |x| + 1 \]

with the conditions \( f(-2) = 5 \) and \( f(1) = 1 \).

### Step 1: Substitute \( x = 2 \) into the equation

We start by substituting \( x = 2 \) into the given equation:

\[ m f(2 - 1) + n f(-2) = 2 |2| + 1 \]

This simplifies to:

\[ m f(1) + n f(-2) = 2 \cdot 2 + 1 \]

### Step 2: Substitute known values

Using the known values \( f(1) = 1 \) and \( f(-2) = 5 \):

\[ m \cdot 1 + n \cdot 5 = 4 + 1 \]

This simplifies to:

\[ m + 5n = 5 \quad \text{(Equation 1)} \]

### Step 3: Substitute \( x = -1 \) into the equation

Next, we substitute \( x = -1 \) into the original equation:

\[ m f(-1 - 1) + n f(-(-1)) = 2 |-1| + 1 \]

This simplifies to:

\[ m f(-2) + n f(1) = 2 \cdot 1 + 1 \]

### Step 4: Substitute known values again

Using \( f(-2) = 5 \) and \( f(1) = 1 \):

\[ m \cdot 5 + n \cdot 1 = 2 + 1 \]

This simplifies to:

\[ 5m + n = 3 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations

Now we have a system of two equations:

1. \( m + 5n = 5 \) (Equation 1)
2. \( 5m + n = 3 \) (Equation 2)

We can solve these equations simultaneously.

From Equation 1, we can express \( m \) in terms of \( n \):

\[ m = 5 - 5n \]

### Step 6: Substitute \( m \) into Equation 2

Substituting \( m \) into Equation 2:

\[ 5(5 - 5n) + n = 3 \]

This simplifies to:

\[ 25 - 25n + n = 3 \]

Combining like terms gives:

\[ 25 - 24n = 3 \]

### Step 7: Solve for \( n \)

Rearranging gives:

\[ -24n = 3 - 25 \]
\[ -24n = -22 \]
\[ n = \frac{22}{24} = \frac{11}{12} \]

### Step 8: Substitute \( n \) back to find \( m \)

Now substitute \( n = \frac{11}{12} \) back into the expression for \( m \):

\[ m = 5 - 5 \left(\frac{11}{12}\right) \]

Calculating this gives:

\[ m = 5 - \frac{55}{12} = \frac{60}{12} - \frac{55}{12} = \frac{5}{12} \]

### Final Answer

Thus, the values of \( m \) and \( n \) are:

\[ m = \frac{5}{12}, \quad n = \frac{11}{12} \]