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

Options

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

Answer

To solve the problem step by step, we start with the function given:

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

### Step 2: Use the first condition \( f(1) = 12 \)
Substituting \( x = 1 \) into the function:
\[ f(1) = a(1^n + 3) = 12 \]
This simplifies to:
\[ a(1 + 3) = 12 \]
\[ 4a = 12 \]
Now, solve for \( a \):
\[ a = \frac{12}{4} = 3 \]

### Step 3: Use the second condition \( f(3) = 36 \)
Now, substituting \( x = 3 \) into the function:
\[ f(3) = a(3^n + 3) = 36 \]
Substituting \( a = 3 \):
\[ 3(3^n + 3) = 36 \]
Dividing both sides by 3:
\[ 3^n + 3 = 12 \]
Now, solve for \( 3^n \):
\[ 3^n = 12 - 3 \]
\[ 3^n = 9 \]
This implies:
\[ 3^n = 3^2 \]
Thus, we find:
\[ n = 2 \]

### Step 4: Write the function with known values of \( a \) and \( n \)
Now we can write the function as:
\[ f(x) = 3(x^2 + 3) \]

### Step 5: Find \( f(2) \)
Now, we need to find \( f(2) \):
\[ f(2) = 3(2^2 + 3) \]
Calculating \( 2^2 \):
\[ f(2) = 3(4 + 3) \]
\[ f(2) = 3(7) \]
Finally:
\[ f(2) = 21 \]

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

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