Use the given functions to find each of the following and their respective domains. Do not try to simplify the resulting functions and be sure to write each domain using interval notation.
f(x)=3/x and g(x)=x-6
(f+g)(x)=
(f-g)(x)=
(fg)(x)=
(f/g)(x)=
what are the domains for all?
To find each of the resulting functions, we perform the operations on f(x) and g(x):
1. (f+g)(x) = f(x) + g(x) = 3/x + x - 6
2. (f-g)(x) = f(x) - g(x) = 3/x - (x - 6) = 3/x - x + 6
3. (fg)(x) = f(x) * g(x) = (3/x)(x - 6) = 3 - 18/x
4. (f/g)(x) = f(x) / g(x) = 3/x / (x - 6) = 3/x * 1/(x - 6) = 3/(x^2 - 6x)
Now, we need to determine the domain of each of the resulting functions:
1. (f+g)(x) = 3/x + x - 6
Domain: x cannot be equal to 0 because of the division by x.
Domain: (-∞, 0) U (0, ∞)
2. (f-g)(x) = 3/x - x + 6
Domain: x cannot be equal to 0 because of the division by x.
Domain: (-∞, 0) U (0, ∞)
3. (fg)(x) = 3 - 18/x
Domain: x cannot be equal to 0 because of the division by x.
Domain: (-∞, 0) U (0, ∞)
4. (f/g)(x) = 3/(x^2 - 6x)
Domain: x^2 - 6x = 0
x(x - 6) = 0
x = 0 or x = 6
Since x cannot be 0 or 6, the domain is all real numbers except 0 and 6.
Domain: (-∞, 0) U (0, 6) U (6, ∞)