If `g(x) = 2x^(2) + 3x - 4` and `g (f(x)) = 8x^(2) + 14 x + 1` then f (2) =

Options

A. 3
B. 5
C. `-3`
D. `-5`

Explanation

To solve the problem step by step, we start with the given functions and the equation we need to solve.

### Step 1: Write down the given functions
We have:
- \( g(x) = 2x^2 + 3x - 4 \)
- \( g(f(x)) = 8x^2 + 14x + 1 \)

### Step 2: Substitute \( f(x) \) into \( g(x) \)
We know that \( g(f(x)) \) can be expressed as:
\[ g(f(x)) = 2(f(x))^2 + 3f(x) - 4 \]
This means we can set up the equation:
\[ 2(f(x))^2 + 3f(x) - 4 = 8x^2 + 14x + 1 \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ 2(f(x))^2 + 3f(x) - 4 - (8x^2 + 14x + 1) = 0 \]
This simplifies to:
\[ 2(f(x))^2 + 3f(x) - 8x^2 - 14x - 5 = 0 \]

### Step 4: Substitute \( x = 2 \)
Now, we need to find \( f(2) \). We substitute \( x = 2 \) into the equation:
\[ 2(f(2))^2 + 3f(2) - 8(2^2) - 14(2) - 5 = 0 \]
Calculating the terms:
\[ 2(f(2))^2 + 3f(2) - 8(4) - 14(2) - 5 = 0 \]
\[ 2(f(2))^2 + 3f(2) - 32 - 28 - 5 = 0 \]
\[ 2(f(2))^2 + 3f(2) - 65 = 0 \]

### Step 5: Let \( f(2) = t \)
Let \( f(2) = t \). The equation becomes:
\[ 2t^2 + 3t - 65 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2 \), \( b = 3 \), and \( c = -65 \).
\[ t = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-65)}}{2 \cdot 2} \]
Calculating the discriminant:
\[ = \frac{-3 \pm \sqrt{9 + 520}}{4} \]
\[ = \frac{-3 \pm \sqrt{529}}{4} \]
\[ = \frac{-3 \pm 23}{4} \]

### Step 7: Calculate the possible values for \( t \)
Calculating the two possible values:
1. \( t = \frac{20}{4} = 5 \)
2. \( t = \frac{-26}{4} = -\frac{13}{2} \)

### Step 8: Conclusion
Thus, the possible values for \( f(2) \) are \( 5 \) and \( -\frac{13}{2} \). However, since we are looking for a function value, we take the positive value:
\[ f(2) = 5 \]

### Final Answer
\[ f(2) = 5 \]