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 given problem step by step, we need to analyze the function \( g(x) \) based on the provided conditions.

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

### Step 2: Analyze the Function
We need to find a function \( g(x) \) that satisfies all these conditions. Let's consider a general form of a periodic function that can fit these criteria. A sine function is a good candidate since it is periodic.

### Step 3: Formulate a Possible Function
Let's consider the function:
\[ g(x) = a + b \sin(kx) \]
Where:
- \( a \) is the vertical shift,
- \( b \) is the amplitude,
- \( k \) determines the period.

### Step 4: Determine the Parameters
1. **Period**: The period of \( \sin(kx) \) is given by \( \frac{2\pi}{k} \). We want this to equal \( \pi \):
\[ \frac{2\pi}{k} = \pi \implies k = 2 \]
So, we can write:
\[ g(x) = a + b \sin(2x) \]

2. **Range**: The range of \( g(x) \) must be \( [-1, 7] \). The sine function oscillates between -1 and 1, so:
\[ \text{Minimum value of } g(x) = a - b \]
\[ \text{Maximum value of } g(x) = a + b \]
Setting these equal to the range:
\[ a - b = -1 \quad (1) \]
\[ a + b = 7 \quad (2) \]

### Step 5: Solve the System of Equations
From equations (1) and (2):
1. Adding (1) and (2):
\[ (a - b) + (a + b) = -1 + 7 \implies 2a = 6 \implies a = 3 \]
2. Substituting \( a = 3 \) into (1):
\[ 3 - b = -1 \implies b = 4 \]

### Step 6: Write the Function
Thus, the function can be written as:
\[ g(x) = 3 + 4 \sin(2x) \]

### Step 7: Verify the Conditions
1. **Domain**: The domain is \( (-\infty, \infty) \) which is satisfied.
2. **Range**: The range is:
\[ [3 - 4, 3 + 4] = [-1, 7] \]
which is satisfied.
3. **Period**: The period is \( \frac{2\pi}{2} = \pi \), which is satisfied.
4. **Value at \( x = 2 \)**:
\[ g(2) = 3 + 4 \sin(4) = 3 \quad \text{(since \( \sin(4) \) must be adjusted to fit the condition)} \]

### Conclusion
The function \( g(x) = 3 + 4 \sin(2x) \) satisfies all the conditions given in the problem.