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PROBLEM

Given f(x)=−3x2+x−7f(x)=-3 x^{2}+x-7 and g(x)=5x+11g(x)=5 x+11, find:
a) (f+g)(x)(f+g)(x)
−3x2+6x+4-3 x^{2}+6 x+4 06
b) the domain, in interval notation, of (f+g)(x)(f+g)(x)
(−∞,∞)↷σ4(-\infty, \infty) \curvearrowright \sigma^{4} c) (f−g)(x)(f-g)(x)
−3x2−4x−18-3 x^{2}-4 x-18 d) the domain, in interval notation, of (f−g)(x)(f-g)(x)
(−∞,∞)6(-\infty, \infty) \sqrt{6} e) (f−g)(x)(f-g)(x)
−15x3−28x2−24x−77σ6-15 x^{3}-28 x^{2}-24 x-77 \quad \sigma^{6} f) the domain, in interval notation, of ( f⋅g)(x)\mathrm{f} \cdot \mathrm{g})(\mathrm{x})
(−∞,∞)↷06(-\infty, \infty) \curvearrowright 0^{6}

STEP 1

1. We are given two functions: f(x)=−3x2+x−7 f(x) = -3x^2 + x - 7 and g(x)=5x+11 g(x) = 5x + 11 .
2. We need to find the sum, difference, and product of these functions, as well as their respective domains.

STEP 2

1. Calculate (f+g)(x)(f+g)(x).
2. Determine the domain of (f+g)(x)(f+g)(x).
3. Calculate (f−g)(x)(f-g)(x).
4. Determine the domain of (f−g)(x)(f-g)(x).
5. Calculate (f⋅g)(x)(f \cdot g)(x).
6. Determine the domain of (f⋅g)(x)(f \cdot g)(x).

STEP 3

Calculate (f+g)(x)(f+g)(x):
(f+g)(x)=f(x)+g(x) (f+g)(x) = f(x) + g(x) =(−3x2+x−7)+(5x+11) = (-3x^2 + x - 7) + (5x + 11)

STEP 4

Simplify the expression:
(f+g)(x)=−3x2+x−7+5x+11 (f+g)(x) = -3x^2 + x - 7 + 5x + 11 =−3x2+6x+4 = -3x^2 + 6x + 4

STEP 5

Determine the domain of (f+g)(x)(f+g)(x):
Since (f+g)(x)=−3x2+6x+4(f+g)(x) = -3x^2 + 6x + 4 is a polynomial, its domain is all real numbers.
Domain: (−∞,∞)(- \infty, \infty)

STEP 6

Calculate (f−g)(x)(f-g)(x):
(f−g)(x)=f(x)−g(x) (f-g)(x) = f(x) - g(x) =(−3x2+x−7)−(5x+11) = (-3x^2 + x - 7) - (5x + 11)

STEP 7

Simplify the expression:
(f−g)(x)=−3x2+x−7−5x−11 (f-g)(x) = -3x^2 + x - 7 - 5x - 11 =−3x2−4x−18 = -3x^2 - 4x - 18

STEP 8

Determine the domain of (f−g)(x)(f-g)(x):
Since (f−g)(x)=−3x2−4x−18(f-g)(x) = -3x^2 - 4x - 18 is a polynomial, its domain is all real numbers.
Domain: (−∞,∞)(- \infty, \infty)

STEP 9

Calculate (f⋅g)(x)(f \cdot g)(x):
(f⋅g)(x)=f(x)⋅g(x) (f \cdot g)(x) = f(x) \cdot g(x) =(−3x2+x−7)(5x+11) = (-3x^2 + x - 7)(5x + 11)

STEP 10

Expand the expression:
(f⋅g)(x)=(−3x2)(5x)+(−3x2)(11)+(x)(5x)+(x)(11)+(−7)(5x)+(−7)(11) (f \cdot g)(x) = (-3x^2)(5x) + (-3x^2)(11) + (x)(5x) + (x)(11) + (-7)(5x) + (-7)(11) =−15x3−33x2+5x2+11x−35x−77 = -15x^3 - 33x^2 + 5x^2 + 11x - 35x - 77 =−15x3−28x2−24x−77 = -15x^3 - 28x^2 - 24x - 77

SOLUTION

Determine the domain of (f⋅g)(x)(f \cdot g)(x):
Since (f⋅g)(x)=−15x3−28x2−24x−77(f \cdot g)(x) = -15x^3 - 28x^2 - 24x - 77 is a polynomial, its domain is all real numbers.
Domain: (−∞,∞)(- \infty, \infty)
The solutions are:
a) (f+g)(x)=−3x2+6x+4(f+g)(x) = -3x^2 + 6x + 4
b) Domain of (f+g)(x)(f+g)(x): (−∞,∞)(- \infty, \infty)
c) (f−g)(x)=−3x2−4x−18(f-g)(x) = -3x^2 - 4x - 18
d) Domain of (f−g)(x)(f-g)(x): (−∞,∞)(- \infty, \infty)
e) (f⋅g)(x)=−15x3−28x2−24x−77(f \cdot g)(x) = -15x^3 - 28x^2 - 24x - 77
f) Domain of (f⋅g)(x)(f \cdot g)(x): (−∞,∞)(- \infty, \infty)

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