If `f:R rarr R` is a function satisfying the equation `f(3x+1)+f(3x+10)=10, Aax in R`, then the period of f(x) is

Answer

To find the period of the function \( f: \mathbb{R} \to \mathbb{R} \) satisfying the equation

\[ f(3x + 1) + f(3x + 10) = 10, \]

we can follow these steps:

### Step 1: Substitute a variable
Let \( t = 3x + 1 \). Then, we can express \( 3x + 10 \) in terms of \( t \):

\[ 3x + 10 = (3x + 1) + 9 = t + 9. \]

### Step 2: Rewrite the equation
Substituting \( t \) into the original equation gives us:

\[ f(t) + f(t + 9) = 10. \]

### Step 3: Create a new equation
Now, we can create another equation by substituting \( t + 9 \) into the function:

\[ f(t + 9) + f(t + 18) = 10. \]

### Step 4: Set up a system of equations
We now have two equations:

1. \( f(t) + f(t + 9) = 10 \) (Equation 1)
2. \( f(t + 9) + f(t + 18) = 10 \) (Equation 2)

### Step 5: Subtract the first equation from the second
Subtract Equation 1 from Equation 2:

\[ (f(t + 9) + f(t + 18)) - (f(t) + f(t + 9)) = 10 - 10. \]

This simplifies to:

\[ f(t + 18) - f(t) = 0. \]

### Step 6: Conclude the periodicity
From the above equation, we see that:

\[ f(t + 18) = f(t). \]

This indicates that the function \( f \) is periodic with a period of 18.

### Final Answer
Thus, the period of the function \( f(x) \) is

\[ \boxed{18}. \]