Let \( f(x)=\sqrt{3 x-1} \) and \( g(x)=\frac{1}{x} \). Find \( (f+g)(x),(f-g)(x),(f g)(x) \), and \( \left(\frac{f}{g}\right)(x) \). Give the domain of each. \( (f+g)(x)=\sqrt{4 x-1}+\frac{1}{x} \) (Simplify your answer.)

To find the expressions for \( (f+g)(x) \), \( (f-g)(x) \), \( (fg)(x) \), and \( \left(\frac{f}{g}\right)(x) \), we start with the given functions: \[ f(x) = \sqrt{3x - 1} \] \[ g(x) = \frac{1}{x} \] ### 1. Finding \( (f+g)(x) \) \[ (f+g)(x) = f(x) + g(x) = \sqrt{3x - 1} + \frac{1}{x} \] **Domain of \( (f+g)(x) \)**: - For \( f(x) \) to be defined, \( 3x - 1 \geq 0 \) which gives \( x \geq \frac{1}{3} \). - For \( g(x) \) to be defined, \( x \neq 0 \). Thus, the domain of \( (f+g)(x) \) is \( x \in \left[\frac{1}{3}, \infty\right) \). ### 2. Finding \( (f-g)(x) \) \[ (f-g)(x) = f(x) - g(x) = \sqrt{3x - 1} - \frac{1}{x} \] **Domain of \( (f-g)(x) \)**: - The same conditions apply as for \( (f+g)(x) \). Thus, the domain of \( (f-g)(x) \) is also \( x \in \left[\frac{1}{3}, \infty\right) \). ### 3. Finding \( (fg)(x) \) \[ (fg)(x) = f(x) \cdot g(x) = \sqrt{3x - 1} \cdot \frac{1}{x} = \frac{\sqrt{3x - 1}}{x} \] **Domain of \( (fg)(x) \)**: - The same conditions apply as for \( (f+g)(x) \). Thus, the domain of \( (fg)(x) \) is also \( x \in \left[\frac{1}{3}, \infty\right) \). ### 4. Finding \( \left(\frac{f}{g}\right)(x) \) \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{3x - 1}}{\frac{1}{x}} = x\sqrt{3x - 1} \] **Domain of \( \left(\frac{f}{g}\right)(x) \)**: - The same conditions apply as for \( (f+g)(x) \). Thus, the domain of \( \left(\frac{f}{g}\right)(x) \) is also \( x \in \left[\frac{1}{3}, \infty\right) \). ### Summary of Results 1. \( (f+g)(x) = \sqrt{3x - 1} + \frac{1}{x} \) with domain \( x \in \left[\frac{1}{3}, \infty\right) \). 2. \( (f-g)(x) = \sqrt{3x - 1} - \frac{1}{x} \) with domain \( x \in \left[\frac{1}{3}, \infty\right) \). 3. \( (fg)(x) = \frac{\sqrt{3x - 1}}{x} \) with domain \( x \in \left[\frac{1}{3}, \infty\right) \). 4. \( \left(\frac{f}{g}\right)(x) = x\sqrt{3x - 1} \) with domain \( x \in \left[\frac{1}{3}, \infty\right) \).