A function `g(x)` satisfies the following conditions
(i) Domain of `g` is `(-oo,oo)` (ii) Range is `g` is `[-1,7]`
(iii) `g` has a period `pi` and (iv) `g(2)=3`
Then which of the following may be possible.

Options

A. `g(x)=3+4sin(npi+2x-4),n epsilonI`
B. `g(x)=[(3,,,x=npi),(3+4sin,,,x!=npi):}`
C. `g(x)=3+4cos(npi+2x-4),n epsilonI`
D. `g(x)=3-8sin(npi+2x-4),n epsilonI`

Explanation

To solve the problem, we need to analyze the function \( g(x) \) based on the given conditions. Let's go through the steps systematically.

### Step 1: Understanding the Conditions
We have the following conditions for the function \( g(x) \):
1. The domain of \( g \) is \( (-\infty, \infty) \).
2. The range of \( g \) is \( [-1, 7] \).
3. \( g \) has a period of \( \pi \).
4. \( g(2) = 3 \).

### Step 2: Analyzing the Function
We need to find a function that meets all these conditions. A common function that has a range and periodicity is a sine or cosine function.

### Step 3: Considering a Candidate Function
Let's consider a function of the form:
\[ g(x) = a + b \sin(kx) \]
where \( a \) is the vertical shift, \( b \) is the amplitude, and \( k \) is related to the period.

### Step 4: Setting the Parameters
1. **Range**: The maximum value of \( g(x) \) will be \( a + b \) and the minimum value will be \( a - b \). To satisfy the range \( [-1, 7] \):
- Set \( a - b = -1 \) (minimum value)
- Set \( a + b = 7 \) (maximum value)

Solving these equations:
\[ a - b = -1 \quad (1) \]
\[ a + b = 7 \quad (2) \]
Adding (1) and (2):
\[ 2a = 6 \implies a = 3 \]
Substituting \( a = 3 \) into (2):
\[ 3 + b = 7 \implies b = 4 \]

2. **Period**: The period \( T \) of the sine function is given by \( T = \frac{2\pi}{k} \). We want \( T = \pi \):
\[ \frac{2\pi}{k} = \pi \implies k = 2 \]

### Step 5: Formulating the Function
Thus, the candidate function can be written as:
\[ g(x) = 3 + 4 \sin(2x) \]

### Step 6: Checking the Conditions
1. **Domain**: The domain of \( g(x) \) is \( (-\infty, \infty) \) since sine is defined for all real numbers.
2. **Range**: The range is \( [3 - 4, 3 + 4] = [-1, 7] \), which satisfies the range condition.
3. **Period**: The period is \( \frac{2\pi}{2} = \pi \), which satisfies the periodicity condition.
4. **Value at \( g(2) \)**:
\[ g(2) = 3 + 4 \sin(2 \cdot 2) = 3 + 4 \sin(4) \]
We need to check if \( \sin(4) \) can yield \( 0 \) (as \( n\pi \) for integer \( n \)):
\[ g(2) = 3 + 4 \cdot 0 = 3 \]

### Conclusion
The function \( g(x) = 3 + 4 \sin(2x) \) satisfies all the given conditions. Thus, the answer is that this function may be a possible candidate.