If `f(x) = a(x^n +3), f(1) = 12, f(3) = 36`, then f(2) is equal to

Options

A. 21
B. 18
C. 24
D. 27

Answer

To solve the problem step by step, we will follow the given information and derive the required values systematically.

### Step 1: Write down the function
Given the function:
\[ f(x) = a(x^n + 3) \]

### Step 2: Use the first condition
We know that:
\[ f(1) = 12 \]
Substituting \( x = 1 \) into the function:
\[ f(1) = a(1^n + 3) = a(1 + 3) = a \cdot 4 \]
Setting this equal to 12:
\[ 4a = 12 \]

### Step 3: Solve for \( a \)
Dividing both sides by 4:
\[ a = \frac{12}{4} = 3 \]

### Step 4: Use the second condition
Now, we know that:
\[ f(3) = 36 \]
Substituting \( x = 3 \) into the function:
\[ f(3) = a(3^n + 3) \]
Substituting \( a = 3 \):
\[ f(3) = 3(3^n + 3) = 36 \]

### Step 5: Simplify the equation
Dividing both sides by 3:
\[ 3^n + 3 = 12 \]
Subtracting 3 from both sides:
\[ 3^n = 12 - 3 = 9 \]

### Step 6: Solve for \( n \)
We know that:
\[ 3^n = 9 \]
Since \( 9 = 3^2 \):
\[ n = 2 \]

### Step 7: Write the final function
Now we have \( a = 3 \) and \( n = 2 \). Therefore, the function becomes:
\[ f(x) = 3(x^2 + 3) \]

### Step 8: Find \( f(2) \)
Now we need to find \( f(2) \):
\[ f(2) = 3(2^2 + 3) \]
Calculating \( 2^2 \):
\[ f(2) = 3(4 + 3) = 3 \cdot 7 = 21 \]

### Final Answer
Thus, the value of \( f(2) \) is:
\[ \boxed{21} \]