perform (f×g)(x) and (f over g)(x) on this problem

4. f(x)=x squared; g(x)=1 over 2 x

step by step clearly and consecutively

To solve the problem where you need to perform the operations \( (f \times g)(x) \) and \( \left(\frac{f}{g}\right)(x) \) given the functions \( f(x) = x^2 \) and \( g(x) = \frac{1}{2x} \), we'll proceed step by step.

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

### 2. **Operation \( (f \times g)(x) \):**
   - The operation \( (f \times g)(x) \) represents the product of the two functions, which is calculated as:
     \[
     (f \times g)(x) = f(x) \times g(x)
     \]
   - Substitute the expressions for \( f(x) \) and \( g(x) \):
     \[
     (f \times g)(x) = x^2 \times \frac{1}{2x}
     \]

   - Simplify the expression by multiplying:
     \[
     (f \times g)(x) = \frac{x^2}{2x} = \frac{x \cdot x}{2 \cdot x} = \frac{x}{2}
     \]

   - Therefore:
     \[
     (f \times g)(x) = \frac{x}{2}
     \]

### 3. **Operation \( \left(\frac{f}{g}\right)(x) \):**
   - The operation \( \left(\frac{f}{g}\right)(x) \) represents the division of the two functions, which is calculated as:
     \[
     \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}
     \]
   - Substitute the expressions for \( f(x) \) and \( g(x) \):
     \[
     \left(\frac{f}{g}\right)(x) = \frac{x^2}{\frac{1}{2x}}
     \]

   - Simplify the expression by multiplying by the reciprocal of the denominator:
     \[
     \left(\frac{f}{g}\right)(x) = x^2 \times \frac{2x}{1} = 2x^3
     \]

   - Therefore:
     \[
     \left(\frac{f}{g}\right)(x) = 2x^3
     \]

### 4. **Final Results:**
   - The product of the functions: \( (f \times g)(x) = \frac{x}{2} \)
   - The quotient of the functions: \( \left(\frac{f}{g}\right)(x) = 2x^3 \)

Would you like more details on any step or further explanations? 

Here are some related questions you might find interesting:
1. What happens to the expressions when \( x = 0 \)?
2. How do you find the domain of \( (f \times g)(x) \) and \( \left(\frac{f}{g}\right)(x) \)?
3. What is the derivative of \( (f \times g)(x) \) and \( \left(\frac{f}{g}\right)(x) \)?
4. How would you graph \( (f \times g)(x) \) and \( \left(\frac{f}{g}\right)(x) \)?
5. Can these functions be simplified further for any specific values of \( x \)?

**Tip:** When multiplying or dividing functions, always simplify the expression by canceling out common factors, if possible, to avoid unnecessary complexity.

Learn how to perform operations (f × g)(x) and (f / g)(x) using the functions f(x) = x^2 and g(x) = 1/(2x). Step-by-step solution provided with detailed calculations and simplifications. Suitable for advanced high school students.

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