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 ( (-infty, infty) ).
(type your answer in interval notation.)

the domain of ( f - g ) is ( square ).
(type your answer in interval notation.)

Explanation:

Step1: Analyze the function \( (f - g)(x) \)

The function \( (f - g)(x) = 5x + 12 \), which is a linear function. Linear functions are polynomials of degree 1, and the domain of a polynomial function (including linear functions) 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 \( x \).

Step2: Write the domain in interval notation

For all real numbers, the interval notation is \( (-\infty, \infty) \).

Answer:

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