If `f` is periodic, `g` is polynomial function and `f(g(x))` is periodic and `g(2)=3,g(4)= 7` then `g(6)` is

Options

A. 13
B. 15
C. 11
D. None of these

Explanation

To solve the problem, we need to find the value of \( g(6) \) given that \( g \) is a polynomial function and we have the values of \( g(2) = 3 \) and \( g(4) = 7 \).

### Step 1: Assume the form of the polynomial
Since we have only two values, we can assume that \( g(x) \) is a linear polynomial of the form:
\[ g(x) = ax + b \]

### Step 2: Set up equations using the given values
Using the given values:
1. For \( g(2) = 3 \):
\[ 2a + b = 3 \quad \text{(Equation 1)} \]
2. For \( g(4) = 7 \):
\[ 4a + b = 7 \quad \text{(Equation 2)} \]

### Step 3: Subtract the equations to eliminate \( b \)
Now, we can subtract Equation 1 from Equation 2 to find \( a \):
\[ (4a + b) - (2a + b) = 7 - 3 \]
This simplifies to:
\[ 2a = 4 \]
Thus, we find:
\[ a = 2 \]

### Step 4: Substitute \( a \) back to find \( b \)
Now, substitute \( a = 2 \) back into Equation 1 to find \( b \):
\[ 2(2) + b = 3 \]
This simplifies to:
\[ 4 + b = 3 \]
So, we find:
\[ b = 3 - 4 = -1 \]

### Step 5: Write the polynomial function
Now we can write the polynomial function:
\[ g(x) = 2x - 1 \]

### Step 6: Calculate \( g(6) \)
Finally, we can find \( g(6) \):
\[ g(6) = 2(6) - 1 = 12 - 1 = 11 \]

### Final Answer
Thus, the value of \( g(6) \) is:
\[ \boxed{11} \]