If `f(x) = 2x` and `g(x) = (x^(2))/(2)+1` , then which of the following can be a discontinuous functions?

Options

A. `f(x)+g(x)`
B. `f(x)-g(x)`
C. `f(x).g(x)`
D. `(g(x))/(f(x))`

Explanation

To determine which of the given functions can be discontinuous, we will analyze each option step by step.

### Given Functions:
- \( f(x) = 2x \)
- \( g(x) = \frac{x^2}{2} + 1 \)

### Options to Analyze:
1. \( A = f(x) + g(x) \)
2. \( B = f(x) - g(x) \)
3. \( C = f(x) \cdot g(x) \)
4. \( D = \frac{g(x)}{f(x)} \)

### Step-by-Step Analysis:

#### Option 1: \( A = f(x) + g(x) \)
1. Calculate \( A \):
\[ A = f(x) + g(x) = 2x + \left(\frac{x^2}{2} + 1\right) = \frac{x^2}{2} + 2x + 1 \]
2. Identify the type of function:
- This is a quadratic function, which is continuous for all \( x \in \mathbb{R} \).

#### Option 2: \( B = f(x) - g(x) \)
1. Calculate \( B \):
\[ B = f(x) - g(x) = 2x - \left(\frac{x^2}{2} + 1\right) = -\frac{x^2}{2} + 2x - 1 \]
2. Identify the type of function:
- This is also a quadratic function, which is continuous for all \( x \in \mathbb{R} \).

#### Option 3: \( C = f(x) \cdot g(x) \)
1. Calculate \( C \):
\[ C = f(x) \cdot g(x) = 2x \cdot \left(\frac{x^2}{2} + 1\right) = 2x \cdot \frac{x^2}{2} + 2x = x^3 + 2x \]
2. Identify the type of function:
- This is a cubic polynomial, which is continuous for all \( x \in \mathbb{R} \).

#### Option 4: \( D = \frac{g(x)}{f(x)} \)
1. Calculate \( D \):
\[ D = \frac{g(x)}{f(x)} = \frac{\frac{x^2}{2} + 1}{2x} \]
Simplifying this gives:
\[ D = \frac{x^2 + 2}{4x} \]
2. Identify points of discontinuity:
- The function \( D \) is undefined when \( f(x) = 0 \), which occurs at \( x = 0 \).
- As \( x \) approaches 0, \( D \) approaches infinity, indicating a discontinuity at \( x = 0 \).

### Conclusion:
The function that can be discontinuous is:
\[ \text{Option } D = \frac{g(x)}{f(x)} \]