Question

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

Explanation:

Step1: Recall the domain of polynomial functions

The functions \(f(x) = 3x + 4\) and \(g(x)=2x - 5\) are both linear (polynomial) functions. The domain of any polynomial function (including linear functions) is all real numbers, which in interval notation is \((-\infty, \infty)\).

Step2: Determine the domain of the product function \((fg)(x)\)

The product function \((fg)(x)=f(x)\cdot g(x)=(3x + 4)(2x - 5)=6x^{2}-7x - 20\), which is also a polynomial function (a quadratic function). Since the domain of a polynomial function is all real numbers, the domain of \((fg)(x)\) is the set of all real numbers. In interval notation, this is \((-\infty, \infty)\).

Answer:

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