If `g(x)=x^2-1` and gof (x)=`x^2+4x+3` , then `f(1/2)` is equal `(f (x) gt 0 AA x in R)` :

Options

A. `3/2`
B. `2`
C. `5/2`
D. None of these

Explanation

To solve the problem, we need to find the value of \( f\left(\frac{1}{2}\right) \) given the functions \( g(x) = x^2 - 1 \) and \( g(f(x)) = x^2 + 4x + 3 \).

### Step-by-Step Solution:

1. **Understand the given functions**:
- We have \( g(x) = x^2 - 1 \).
- We also know that \( g(f(x)) = x^2 + 4x + 3 \).

2. **Substituting \( f(x) \) into \( g(x) \)**:
- From the definition of \( g(f(x)) \), we can write:
\[ g(f(x)) = f(x)^2 - 1 \]
- Therefore, we can equate this to the given expression:
\[ f(x)^2 - 1 = x^2 + 4x + 3 \]

3. **Rearranging the equation**:
- Adding 1 to both sides gives:
\[ f(x)^2 = x^2 + 4x + 4 \]
- This can be factored as:
\[ f(x)^2 = (x + 2)^2 \]

4. **Taking the square root**:
- Taking the square root of both sides, we have:
\[ f(x) = x + 2 \quad \text{or} \quad f(x) = -(x + 2) \]
- Since we are looking for \( f\left(\frac{1}{2}\right) \), we will consider the positive root (as it is common in such problems):
\[ f(x) = x + 2 \]

5. **Finding \( f\left(\frac{1}{2}\right) \)**:
- Now, substituting \( x = \frac{1}{2} \):
\[ f\left(\frac{1}{2}\right) = \frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2} \]

### Final Answer:
Thus, \( f\left(\frac{1}{2}\right) = \frac{5}{2} \).

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