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)=-3x^2-9x and g(x)=9x+4
(f+g)(x)=
d=
(f-g)(x)=
d=
(fg)(x)=
d=
(f/g)(x)=
d=
To find (f+g)(x), we need to add the functions f(x) and g(x):
(f+g)(x) = f(x) + g(x)
= (-3x^2 - 9x) + (9x + 4)
= -3x^2 - 9x + 9x + 4
= -3x^2 + 4
The domain of (f+g)(x) is the same as the domain of f(x) and g(x). Since there are no restrictions in the given functions, the domain is all real numbers. Therefore,
(f+g)(x) = -3x^2 + 4
d = (-∞, ∞)
To find (f-g)(x), we need to subtract the function g(x) from f(x):
(f-g)(x) = f(x) - g(x)
= (-3x^2 - 9x) - (9x + 4)
= -3x^2 - 9x - 9x - 4
= -3x^2 - 18x - 4
The domain of (f-g)(x) is the same as the domain of f(x) and g(x), which is all real numbers. Therefore,
(f-g)(x) = -3x^2 - 18x - 4
d = (-∞, ∞)
To find (fg)(x), we need to multiply f(x) and g(x):
(fg)(x) = f(x) * g(x)
= (-3x^2 - 9x) * (9x + 4)
= -3x^2(9x + 4) - 9x(9x + 4)
= -27x^3 - 12x^2 - 81x^2 - 36x
= -27x^3 - 93x^2 - 36x
The domain of (fg)(x) is the same as the domain of f(x) and g(x), which is all real numbers. Therefore,
(fg)(x) = -27x^3 - 93x^2 - 36x
d = (-∞, ∞)
To find (f/g)(x), we need to divide f(x) by g(x):
(f/g)(x) = f(x) / g(x)
= (-3x^2 - 9x) / (9x + 4)
The domain of (f/g)(x) is the same as the domain of f(x) and g(x), which is all real numbers except for values that make the denominator equal to zero. Therefore, x cannot be -4/9 since it would result in division by zero. So, the domain is:
d = (-∞, -4/9) U (-4/9, ∞)
To find each of the following combinations of the given functions, we will substitute the respective functions into the given operations.
1. (f+g)(x):
To find (f+g)(x), we need to add the functions f(x) and g(x).
(f+g)(x) = f(x) + g(x) = (-3x^2-9x) + (9x+4) = -3x^2 + (9x - 9x) + 4 = -3x^2 + 4
Domain (d):
Since f(x) and g(x) are both polynomials, there are no restrictions on the domain. Therefore, the domain of (f+g)(x) is all real numbers or (-∞, ∞) in interval notation.
2. (f-g)(x):
To find (f-g)(x), we need to subtract the function g(x) from f(x).
(f-g)(x) = f(x) - g(x) = (-3x^2-9x) - (9x+4) = -3x^2 - 9x - 9x - 4 = -3x^2 - 18x - 4
Domain (d):
Similar to the addition operation, there are no restrictions on the domain for subtraction. Therefore, the domain of (f-g)(x) is also all real numbers or (-∞, ∞) in interval notation.
3. (fg)(x):
To find (fg)(x), we need to multiply the functions f(x) and g(x).
(fg)(x) = f(x) * g(x) = (-3x^2-9x) * (9x+4) = -27x^3 - 12x^2 - 81x^2 - 36x = -27x^3 - 93x^2 - 36x
Domain (d):
There are no restrictions on the domain for the multiplication operation as well. Therefore, the domain of (fg)(x) is all real numbers or (-∞, ∞) in interval notation.
4. (f/g)(x):
To find (f/g)(x), we need to divide the function f(x) by g(x).
(f/g)(x) = f(x) / g(x) = (-3x^2-9x) / (9x+4)
Domain (d):
The division operation requires that the numerator is defined and the denominator is not equal to zero. Therefore, we need to find the domain where the denominator, 9x+4, is not equal to zero.
9x + 4 ≠ 0
Solving for x:
9x ≠ -4
x ≠ -4/9
Therefore, the domain of (f/g)(x) is all real numbers except x = -4/9 or (-∞, -4/9) U (-4/9, ∞) in interval notation.
To find each of the given functions and their respective domains using the given functions f(x) and g(x), follow these steps:
1. (f+g)(x):
To find (f+g)(x), add the two functions f(x) and g(x) together:
(f+g)(x) = f(x) + g(x) = (-3x^2-9x) + (9x+4)
2. (f-g)(x):
To find (f-g)(x), subtract g(x) from f(x):
(f-g)(x) = f(x) - g(x) = (-3x^2-9x) - (9x+4)
3. (fg)(x):
To find (fg)(x), multiply f(x) and g(x) together:
(fg)(x) = f(x) * g(x) = (-3x^2-9x) * (9x+4)
4. (f/g)(x):
To find (f/g)(x), divide f(x) by g(x):
(f/g)(x) = f(x) / g(x) = (-3x^2-9x) / (9x+4)
Now, let's find the respective domains for each function:
Domain of (f+g)(x):
The domain is the set of all valid input values for which the function is defined. Since addition, subtraction, multiplication, and division of real numbers is defined for all values of x, the domain of (f+g)(x) is the set of all real numbers, which can be represented as (-∞, +∞) in interval notation.
Domain of (f-g)(x):
Like in the case of addition, subtraction is also defined for all real numbers. Therefore, the domain of (f-g)(x) is the set of all real numbers, expressed as (-∞, +∞) in interval notation.
Domain of (fg)(x):
Again, multiplication is defined for all real numbers, so the domain of (fg)(x) is also (-∞, +∞), which represents the set of all real numbers.
Domain of (f/g)(x):
Division, however, has a limitation. We need to consider that division by zero is undefined. Therefore, we need to find the values of x for which g(x) is not equal to zero. From the given g(x) = 9x+4, we can see that g(x) ≠ 0 for any value of x. So, the domain of (f/g)(x) is also (-∞, +∞), representing all real numbers.
In summary, the domains for all the given functions are (-∞, +∞), and the functions themselves are as follows:
(f+g)(x) = (-3x^2-9x) + (9x+4)
(f-g)(x) = (-3x^2-9x) - (9x+4)
(fg)(x) = (-3x^2-9x) * (9x+4)
(f/g)(x) = (-3x^2-9x) / (9x+4)