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)\\)

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

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

(fg)(x) = 14x² - 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 (-∞, ∞).
(type your answer in interval notation.)

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

the domain of fg is
(type your answer in interval notation.)

Explanation:

Step1: Analyze the function (fg)(x)

The function \((fg)(x)\) is the product of \(f(x) = 7x + 5\) and \(g(x) = 2x - 7\). Since both \(f(x)\) and \(g(x)\) are linear functions (polynomials of degree 1), their product \((fg)(x)=14x^{2}-39x - 35\) is a quadratic function (a polynomial of degree 2).

Step2: Determine the domain of a polynomial function

The domain of a polynomial function (including linear, quadratic, cubic, etc.) is all real numbers because there are no restrictions (such as division by zero or square roots of negative numbers) that would exclude any real number \(x\) from being in the domain. For a quadratic function \(y = ax^{2}+bx + c\) (where \(a
eq0\)), we can plug in any real number \(x\) and get a real number \(y\). So, the domain of \((fg)(x)\) (a quadratic polynomial) is all real numbers.

In interval notation, all real numbers are represented as \((-\infty, \infty)\).

Answer:

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