Find f + g, f - g, fg, and f/g and their domains. f(x) = 4 - x, g(x) = x² - 3x

Solution:

If f and g are any two real functions then f + g, f - g fg, and f/g are called the algebra of the functions. Where we add, subtract, multiply and divide the function .We may note that the domain of the such functions will be intersection of the domains of the two functions

If f and g are any two functions then (f+ g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x)

And (f × g)(x) = f(x) × g(x) and given that f(x) = 4 - x and g(x) = x² - 3x

Thus, (f + g) (x) = f(x) + g(x) = 4 - x + x² - 3x = x² - 4x + 4

(f - g)(x) = f(x) - g(x) = 4 - x - (x² - 3x) = 4 + 2x - x²

(f × g) (x) = f(x) × g(x) = ( 4 - x) ( x² - 3x) = 4x² - 12x - x³ + 3x² =- x³ +7x² - 12x

[f/g] (x) = f(x) / g(x) = (4 - x) / (x² - 3x)