Questions: For the given functions f and g, complete parts (a)-(h). For parts (a)-(d), also find the domain. f(x)=3x+4; g(x)=5x-9 (d) Find (t/g)(x). (f/g)(x)=(3x+4)/(5x-9) (Simplify your answer.) What is the domain of f/g? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain is x . (Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The domain is x x is any real number.

Solution Steps

To find \(\left(\frac{f}{g}\right)(x)\), we need to divide the function \(f(x)\) by \(g(x)\). The domain of \(\frac{f}{g}\) is all real numbers except where \(g(x) = 0\), since division by zero is undefined. We will solve for \(x\) where \(g(x) = 0\) to find the values to exclude from the domain.

We have the functions defined as: \[ f(x) = 3x + 4 \quad \text{and} \quad g(x) = 5x - 9 \] To find \(\left(\frac{f}{g}\right)(x)\), we compute: \[ \left(\frac{f}{g}\right)(x) = \frac{3x + 4}{5x - 9} \]

The domain of \(\frac{f}{g}\) is all real numbers except where \(g(x) = 0\). We solve for \(x\) in the equation: \[ 5x - 9 = 0 \] Solving this gives: \[ 5x = 9 \quad \Rightarrow \quad x = \frac{9}{5} \] Thus, the domain of \(\frac{f}{g}\) is all real numbers except \(x = \frac{9}{5}\).

Final Answer