Question

let f(x) = 3x + 4 and g(x) = 2x - 5. find (f + g)(x), (f - g)(x), (fg)(x), and \\(\left(\frac{f}{g}\
ight)(x)\\). give the domain of each.\
(f + g)(x) = 5x - 1 (simplify your answer.)\
(f - g)(x) = x + 9 (simplify your answer.)\
(fg)(x) = 6x² - 7x - 20 (simplify your answer.)\
\\(\left(\frac{f}{g}\
ight)(x) = \frac{3x + 4}{2x - 5}\\) (simplify your answer.)\
the domain of f + g is \\(\square\\).\
(type your answer in interval notation.)

Explanation:

Step1: Recall the domain of linear functions

The functions \(f(x) = 3x + 4\) and \(g(x)=2x - 5\) are both linear functions (polynomial functions of degree 1). The domain of a linear function (or any polynomial function) is all real numbers because there are no restrictions (like division by zero or square roots of negative numbers) that would exclude any real number \(x\) from being in the domain.

Step2: Determine the domain of \(f + g\)

The sum of two functions \(f+g\) is defined as \((f + g)(x)=f(x)+g(x)\). Since the domain of \(f\) is all real numbers (\((-\infty, \infty)\)) and the domain of \(g\) is all real numbers (\((-\infty, \infty)\)), the domain of \(f + g\) is the intersection of the domains of \(f\) and \(g\). The intersection of \((-\infty, \infty)\) with itself is \((-\infty, \infty)\).

Answer:

\((-\infty, \infty)\)