\& Find \( f+g, f-g, f g \) and \( \frac{f}{g} \). Determine the domain for each function. \( f(x)=5 x+2, g(x)=x-6 \) \( (f+g)(x)=6 x-4 \) (Simplify your answer.) What is the domain of \( f+g \) ? A. The domain of \( f+g \) is \( (-\infty, \infty) \). The dype your answer in interval notation.) \( f+g \) is \( \} \). (Use a comma to separate answers as needed.) C. The domain of \( f+g \) is \( \varnothing \). \( (f-g)(x)=\square \) (Simplify your answer.)

Let's find the functions \( f + g \), \( f - g \), \( fg \), and \( \frac{f}{g} \), along with their respective domains. Given: \[ f(x) = 5x + 2 \] \[ g(x) = x - 6 \] ### 1. \( (f + g)(x) \) **Calculation:** \[ \begin{align*} (f + g)(x) &= f(x) + g(x) \\ &= (5x + 2) + (x - 6) \\ &= 6x - 4 \end{align*} \] **Domain:** Both \( f(x) \) and \( g(x) \) are linear functions defined for all real numbers. Therefore, their sum is also defined for all real numbers. \[ \text{Domain of } f + g: \quad (-\infty, \infty) \] ### 2. \( (f - g)(x) \) **Calculation:** \[ \begin{align*} (f - g)(x) &= f(x) - g(x) \\ &= (5x + 2) - (x - 6) \\ &= 4x + 8 \end{align*} \] **Domain:** Similar to the sum, both \( f(x) \) and \( g(x) \) are defined for all real numbers. Thus, their difference is also defined for all real numbers. \[ \text{Domain of } f - g: \quad (-\infty, \infty) \] ### 3. \( (fg)(x) \) **Calculation:** \[ \begin{align*} (fg)(x) &= f(x) \cdot g(x) \\ &= (5x + 2)(x - 6) \\ &= 5x^2 - 30x + 2x - 12 \\ &= 5x^2 - 28x - 12 \end{align*} \] **Domain:** Since \( f(x) \) and \( g(x) \) are polynomials, their product is also a polynomial and is defined for all real numbers. \[ \text{Domain of } fg: \quad (-\infty, \infty) \] ### 4. \( \left(\frac{f}{g}\right)(x) \) **Calculation:** \[ \left(\frac{f}{g}\right)(x) = \frac{5x + 2}{x - 6} \] **Domain:** The function \( \frac{f}{g} \) is defined for all real numbers except where the denominator is zero. Set the denominator equal to zero and solve for \( x \): \[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \] Therefore, the domain excludes \( x = 6 \). \[ \text{Domain of } \frac{f}{g}: \quad (-\infty, 6) \cup (6, \infty) \] --- ### Summary 1. **\( f + g \)** - **Expression:** \( 6x - 4 \) - **Domain:** \( (-\infty, \infty) \) 2. **\( f - g \)** - **Expression:** \( 4x + 8 \) - **Domain:** \( (-\infty, \infty) \) 3. **\( fg \)** - **Expression:** \( 5x^2 - 28x - 12 \) - **Domain:** \( (-\infty, \infty) \) 4. **\( \frac{f}{g} \)** - **Expression:** \( \frac{5x + 2}{x - 6} \) - **Domain:** \( (-\infty, 6) \cup (6, \infty) \)