The functions \( f \) and \( g \) are defined as \( f(x)=4 x-3 \) and \( g(x)=-9 x^{2} \). a) Find the domain of \( f, g, f+g, f-g, f g, f f, \frac{f}{g} \), and \( \frac{g}{f} \). b) Find \( (f+g)(x),(f-g)(x),(f g)(x),(f f)(x),\left(\frac{f}{g}\right)(x) \), and \( \left(\frac{g}{f}\right)(x) \). a) The domain of \( f \) is \( (-\infty, \infty) \). (Type your answer in interval notation.) The domain of \( g \) is \( (-\infty, \infty) \). (Type your answer in interval notation.) The domain of \( f+g \) is (Type your answer in interval notation.)

To find the domains of the functions \( f \) and \( g \), as well as their combinations, we will analyze each function and operation step by step. ### a) Finding the domains 1. **Domain of \( f(x) = 4x - 3 \)**: - This is a linear function, which is defined for all real numbers. - **Domain of \( f \)**: \( (-\infty, \infty) \) 2. **Domain of \( g(x) = -9x^2 \)**: - This is a quadratic function, which is also defined for all real numbers. - **Domain of \( g \)**: \( (-\infty, \infty) \) 3. **Domain of \( f + g \)**: - Since both \( f \) and \( g \) are defined for all real numbers, their sum is also defined for all real numbers. - **Domain of \( f + g \)**: \( (-\infty, \infty) \) 4. **Domain of \( f - g \)**: - Similar to the sum, the difference of two functions defined for all real numbers is also defined for all real numbers. - **Domain of \( f - g \)**: \( (-\infty, \infty) \) 5. **Domain of \( fg \)**: - The product of two functions defined for all real numbers is also defined for all real numbers. - **Domain of \( fg \)**: \( (-\infty, \infty) \) 6. **Domain of \( ff \)**: - This is simply \( f(f(x)) \), which is defined for all real numbers since \( f \) is defined for all real numbers. - **Domain of \( ff \)**: \( (-\infty, \infty) \) 7. **Domain of \( \frac{f}{g} \)**: - The quotient is defined wherever \( g(x) \neq 0 \). We need to find where \( g(x) = -9x^2 = 0 \). - This occurs at \( x = 0 \). Therefore, the domain excludes \( x = 0 \). - **Domain of \( \frac{f}{g} \)**: \( (-\infty, 0) \cup (0, \infty) \) 8. **Domain of \( \frac{g}{f} \)**: - The quotient is defined wherever \( f(x) \neq 0 \). We need to find where \( f(x) = 4x - 3 = 0 \). - This occurs at \( x = \frac{3}{4} \). Therefore, the domain excludes \( x = \frac{3}{4} \). - **Domain of \( \frac{g}{f} \)**: \( (-\infty, \frac{3}{4}) \cup (\frac{3}{4}, \infty) \) ### Summary of Domains - Domain of \( f \): \( (-\infty, \infty) \) - Domain of \( g \): \( (-\infty, \infty) \) - Domain of \( f + g \): \( (-\infty, \infty) \) - Domain of \( f - g \): \( (-\infty, \infty) \) - Domain of \( fg \): \( (-\infty, \infty) \) - Domain of \( ff \): \( (-\infty, \infty) \) - Domain of \( \frac{f}{g} \): \( (-\infty, 0) \cup (0, \infty) \) - Domain of \( \frac{g}{f} \): \( (-\infty, \frac{3}{4}) \cup (\frac{3}{4}, \infty) \) ### b) Finding the functions 1. **Finding \( (f + g)(x) \)**: \[ (f + g)(x) = f(x) + g(x) = (4x - 3) + (-9x^2) = -9x^2 + 4x - 3 \] 2. **Finding \( (f - g)(x) \)**: \[ (f - g)(x) = f(x) - g(x) = (4x - 3) - (-9x^2) = 9x^2 + 4x - 3 \] 3. **Finding \( (fg)(x) \)**: \[ (fg)(x) =