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

To solve the problem, we will go through each part step by step. ### Part a: Finding the Domain 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 also a polynomial function, which is 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 reasoning as above; both functions are defined for all real numbers. - **Domain of \( f - g \):** \( (-\infty, \infty) \) 5. **Domain of \( fg \)**: - The product of two functions that are both defined for all real numbers is also defined for all real numbers. - **Domain of \( fg \):** \( (-\infty, \infty) \) 6. **Domain of \( \frac{f}{g} \)**: - This function is defined wherever \( g(x) \neq 0 \). We need to find when \( g(x) = -9x^2 = 0 \). - This occurs when \( x = 0 \). - **Domain of \( \frac{f}{g} \):** \( (-\infty, 0) \cup (0, \infty) \) 7. **Domain of \( \frac{g}{f} \)**: - This function is defined wherever \( f(x) \neq 0 \). We need to find when \( f(x) = 4x - 3 = 0 \). - This occurs when \( x = \frac{3}{4} \). - **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 \( \frac{f}{g} \):** \( (-\infty, 0) \cup (0, \infty) \) - **Domain of \( \frac{g}{f} \):** \( (-\infty, \frac{3}{4}) \cup (\frac{3}{4}, \infty) \) ### Part 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 \( (f - g)(x) \)**: \[ (f - g)(x) = f(x) - g(x) = (4x - 3) - (-9x^2) = 4x - 3 + 9x^2 = 9x^2 + 4x - 3 \] 3. **Finding \( (fg)(x) \)**: \[ (fg)(x) = f(x) \cdot g(x) = (4x - 3)(-9x^2) \] Expanding this: \[ (fg)(x) = -36x^3 + 27x^2 \] 4. **Finding \( (ff)(x) \)**: \[ (ff)(x) = f(f(x)) = f(4x - 3) = 4(4x - 3) - 3 =