Question

let $f(x) = 7x + 5$ and $g(x) = 2x - 7$. find $(f + g)(x)$, $(f - g)(x)$, $(fg)(x)$, and $left(\frac{f}{g}\
ight)(x)$.

$(f + g)(x) = 9x - 2$ (simplify your answer.)

$(f - g)(x) = 5x + 12$ (simplify your answer.)

$(fg)(x) = 14x^2 - 39x - 35$ (simplify your answer.)

$left(\frac{f}{g}\
ight)(x) = \frac{7x + 5}{2x - 7}$ (simplify your answer.)

the domain of $f + g$ is \boxed{}.
(type your answer in interval notation.)

Explanation:

Step1: Recall domain of sum of functions

The domain of \((f + g)(x)\) is the intersection of the domains of \(f(x)\) and \(g(x)\). Both \(f(x)=7x + 5\) and \(g(x)=2x-7\) are linear functions (polynomial functions of degree 1). The domain of a linear function is all real numbers.

Step2: Determine intersection of domains

The domain of \(f(x)\) is \((-\infty,\infty)\) and the domain of \(g(x)\) is \((-\infty,\infty)\). The intersection of \((-\infty,\infty)\) and \((-\infty,\infty)\) is \((-\infty,\infty)\).

Answer:

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