Questions: Suppose that the functions (f) and (g) are defined as follows. [ f(x)=sqrt3 x+4 ] [ g(x)=-2 x^2+x ] Find (f cdot g) and (f-g). Then, give their domains using interval notation. [ (f cdot g)(x)= ] Domain of (f cdot g) : [ (f-g)(x)= ] Domain of (f-g) :

Solution Steps

To solve this problem, we need to perform two operations on the given functions \( f(x) \) and \( g(x) \): multiplication and subtraction. For each operation, we will also determine the domain of the resulting function.

1. Multiplication \( f \cdot g \): Multiply the expressions for \( f(x) \) and \( g(x) \). The domain of \( f \cdot g \) is the intersection of the domains of \( f \) and \( g \).

2. Subtraction \( f - g \): Subtract the expression for \( g(x) \) from \( f(x) \). The domain of \( f - g \) is also the intersection of the domains of \( f \) and \( g \).

3. Determine Domains:

   • The domain of \( f(x) = \sqrt{3x + 4} \) is where the expression under the square root is non-negative, i.e., \( 3x + 4 \geq 0 \).
   • The domain of \( g(x) = -2x^2 + x \) is all real numbers since it is a polynomial.

The product of the functions \( f \) and \( g \) is given by: \[ (f \cdot g)(x) = \sqrt{3x + 4} \cdot (-2x^2 + x) \]

The difference between the functions \( f \) and \( g \) is expressed as: \[ (f - g)(x) = \sqrt{3x + 4} - (-2x^2 + x) = 2x^2 - x + \sqrt{3x + 4} \]

The domain of \( f(x) = \sqrt{3x + 4} \) is found by ensuring the expression under the square root is non-negative: \[ 3x + 4 \geq 0 \implies x \geq -\frac{4}{3} \] The domain of \( g(x) = -2x^2 + x \) is all real numbers, \( (-\infty, \infty) \). Therefore, the domain of both \( f \cdot g \) and \( f - g \) is the intersection of their domains: \[ \text{Domain of } f \cdot g: \quad \left[-\frac{4}{3}, \infty\right) \] \[ \text{Domain of } f - g: \quad \left[-\frac{4}{3}, \infty\right) \]

Final Answer