Find the following for the given functions. \[ f(x)=\frac{3}{x}, g(x)=\frac{9}{x^{2}-1} \] (a) \( (f+g)(x)=\square \) (b) \( (f-g)(x)=\square \) (c) \( (f g)(x)=\square \) (d) \( (\xi / g)(x)=\square \) What is the domain of \( f / g \) ? (Enter your answer using interval notation.) Need Help? \( \quad \) Road it

To find the expressions for \( (f+g)(x) \), \( (f-g)(x) \), \( (fg)(x) \), and \( \left(\frac{f}{g}\right)(x) \), we will first write down the functions: \[ f(x) = \frac{3}{x}, \quad g(x) = \frac{9}{x^2 - 1} \] ### (a) \( (f+g)(x) \) To find \( (f+g)(x) \): \[ (f+g)(x) = f(x) + g(x) = \frac{3}{x} + \frac{9}{x^2 - 1} \] To combine these fractions, we need a common denominator, which is \( x(x^2 - 1) \): \[ (f+g)(x) = \frac{3(x^2 - 1) + 9x}{x(x^2 - 1)} = \frac{3x^2 - 3 + 9x}{x(x^2 - 1)} = \frac{3x^2 + 9x - 3}{x(x^2 - 1)} \] ### (b) \( (f-g)(x) \) To find \( (f-g)(x) \): \[ (f-g)(x) = f(x) - g(x) = \frac{3}{x} - \frac{9}{x^2 - 1} \] Using the same common denominator \( x(x^2 - 1) \): \[ (f-g)(x) = \frac{3(x^2 - 1) - 9x}{x(x^2 - 1)} = \frac{3x^2 - 3 - 9x}{x(x^2 - 1)} = \frac{3x^2 - 9x - 3}{x(x^2 - 1)} \] ### (c) \( (fg)(x) \) To find \( (fg)(x) \): \[ (fg)(x) = f(x) \cdot g(x) = \frac{3}{x} \cdot \frac{9}{x^2 - 1} = \frac{27}{x(x^2 - 1)} \] ### (d) \( \left(\frac{f}{g}\right)(x) \) To find \( \left(\frac{f}{g}\right)(x) \): \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\frac{3}{x}}{\frac{9}{x^2 - 1}} = \frac{3}{x} \cdot \frac{x^2 - 1}{9} = \frac{3(x^2 - 1)}{9x} = \frac{x^2 - 1}{3x} \] ### Domain of \( \frac{f}{g} \) To find the domain of \( \frac{f}{g} \), we need to consider where \( g(x) \) is not equal to zero and where \( f(x) \) is defined. 1. \( g(x) = \frac{9}{x^2 - 1} \) is undefined when \( x^2 - 1 = 0 \), which gives \( x = 1 \) and \( x = -1 \). 2. \( f(x) = \frac{3}{x} \) is undefined when \( x = 0 \). Thus, the values that are excluded from the domain are \( x = -1, 0, 1 \). The domain in interval notation is: \[ (-\infty, -1) \cup (-1, 0) \cup (0, 1) \cup (1, \infty) \] ### Summary of Results (a) \( (f+g)(x) = \frac{3x^2 + 9x - 3}{x(x^2 - 1)} \) (b) \( (f-g)(x) = \frac{3x^2 - 9x - 3}{x(x^2 - 1)} \) (c) \( (fg)(x) = \frac{27}{x(x^2 - 1)} \) (d) \( \left(\frac{f}{g}\right)(x) = \frac{x^2 - 1}{3x} \) Domain of \( \frac{f}{g} \): \( (-\infty, -1) \cup (-1, 0