Let `f(x)=sqrt(x)`and `g(x) = x`be two functions defined over the set of nonnegative real numbers. Find `(f + g) (x)`, `(f g) (x)`, `(f-g) (x)`and `(f/g)(x)`.

Options

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

Answer

To solve the problem, we need to find the following expressions based on the functions \( f(x) = \sqrt{x} \) and \( g(x) = x \):

1. \( (f + g)(x) \)
2. \( (f \cdot g)(x) \)
3. \( (f - g)(x) \)
4. \( \left( \frac{f}{g} \right)(x) \)

Let's go through each step:

### Step 1: Calculate \( (f + g)(x) \)

\[ (f + g)(x) = f(x) + g(x) \]
\[ = \sqrt{x} + x \]

### Step 2: Calculate \( (f \cdot g)(x) \)

\[ (f \cdot g)(x) = f(x) \cdot g(x) \]
\[ = \sqrt{x} \cdot x \]
\[ = x^{1/2} \cdot x^{1} = x^{(1/2 + 1)} = x^{3/2} \]

### Step 3: Calculate \( (f - g)(x) \)

\[ (f - g)(x) = f(x) - g(x) \]
\[ = \sqrt{x} - x \]

### Step 4: Calculate \( \left( \frac{f}{g} \right)(x) \)

\[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \]
\[ = \frac{\sqrt{x}}{x} \]
\[ = \frac{1}{\sqrt{x}} \quad \text{(for } x \neq 0\text{)} \]

### Summary of Results

1. \( (f + g)(x) = \sqrt{x} + x \)
2. \( (f \cdot g)(x) = x^{3/2} \)
3. \( (f - g)(x) = \sqrt{x} - x \)
4. \( \left( \frac{f}{g} \right)(x) = \frac{1}{\sqrt{x}} \) (for \( x \neq 0 \))