Math Problem Statement

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

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

step by step clearly and consecutively

Solution

To solve the problem where you need to perform the operations (f×g)(x)(f \times g)(x) and (fg)(x)\left(\frac{f}{g}\right)(x) given the functions f(x)=x2f(x) = x^2 and g(x)=12xg(x) = \frac{1}{2x}, we'll proceed step by step.

1. Given Functions:

  • f(x)=x2f(x) = x^2
  • g(x)=12xg(x) = \frac{1}{2x}

2. Operation (f×g)(x)(f \times g)(x):

  • The operation (f×g)(x)(f \times g)(x) represents the product of the two functions, which is calculated as: (f×g)(x)=f(x)×g(x)(f \times g)(x) = f(x) \times g(x)

  • Substitute the expressions for f(x)f(x) and g(x)g(x): (f×g)(x)=x2×12x(f \times g)(x) = x^2 \times \frac{1}{2x}

  • Simplify the expression by multiplying: (f×g)(x)=x22x=xx2x=x2(f \times g)(x) = \frac{x^2}{2x} = \frac{x \cdot x}{2 \cdot x} = \frac{x}{2}

  • Therefore: (f×g)(x)=x2(f \times g)(x) = \frac{x}{2}

3. Operation (fg)(x)\left(\frac{f}{g}\right)(x):

  • The operation (fg)(x)\left(\frac{f}{g}\right)(x) represents the division of the two functions, which is calculated as: (fg)(x)=f(x)g(x)\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}

  • Substitute the expressions for f(x)f(x) and g(x)g(x): (fg)(x)=x212x\left(\frac{f}{g}\right)(x) = \frac{x^2}{\frac{1}{2x}}

  • Simplify the expression by multiplying by the reciprocal of the denominator: (fg)(x)=x2×2x1=2x3\left(\frac{f}{g}\right)(x) = x^2 \times \frac{2x}{1} = 2x^3

  • Therefore: (fg)(x)=2x3\left(\frac{f}{g}\right)(x) = 2x^3

4. Final Results:

  • The product of the functions: (f×g)(x)=x2(f \times g)(x) = \frac{x}{2}
  • The quotient of the functions: (fg)(x)=2x3\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=0x = 0?
  2. How do you find the domain of (f×g)(x)(f \times g)(x) and (fg)(x)\left(\frac{f}{g}\right)(x)?
  3. What is the derivative of (f×g)(x)(f \times g)(x) and (fg)(x)\left(\frac{f}{g}\right)(x)?
  4. How would you graph (f×g)(x)(f \times g)(x) and (fg)(x)\left(\frac{f}{g}\right)(x)?
  5. Can these functions be simplified further for any specific values of xx?

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