Final answer:
Operations on functions f(x) and g(x) include addition, subtraction, multiplication, and division. The resulting functions are obtained by applying arithmetic operations to f(x) and g(x), and for division, domain restrictions must be stated where the denominator cannot be zero.
Step-by-step explanation:
To perform the operations on the functions f(x) = 3x - 7 and g(x) = x² + 3x - 4, we use the definitions of function addition, subtraction, multiplication, and division.
- (f+9)(x): This means we add 9 to the function f(x). So, (f+9)(x) = f(x) + 9 = (3x - 7) + 9 = 3x + 2.
- (f-9)(x): This means we subtract 9 from the function f(x). So, (f-9)(x) = f(x) - 9 = (3x - 7) - 9 = 3x - 16.
- (f · g)(x): This is the product of f(x) and g(x). We multiply the functions together: (f · g)(x) = f(x)g(x) = (3x - 7)(x² + 3x - 4).
- (f/g)(x): This is the division of f(x) by g(x), where g(x) is not equal to 0. We divide the functions: (f/g)(x) = f(x)/g(x) = (3x - 7)/(x² + 3x - 4).
- (g/f)(x): This is the division of g(x) by f(x), where f(x) is not equal to 0. We divide the functions: (g/f)(x) = g(x)/f(x) = (x² + 3x - 4)/(3x - 7).