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 two given functions together:
(f+g)(x) = f(x) + g(x) = (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), which is all real numbers. So, the domain is (-∞, ∞).
To find (f-g)(x), we need to subtract g(x) from f(x):
(f-g)(x) = f(x) - g(x) = (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. So, the domain is (-∞, ∞).
To find (fg)(x), we need to multiply the two given functions together:
(fg)(x) = f(x) * g(x) = (3x^2-9x) * (9x+4) = 27x^3 -3x^2 - 36x^2 - 36x
The domain of (fg)(x) is the same as the domain of f(x) and g(x), which is all real numbers. So, the domain is (-∞, ∞).
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 all real numbers except for any values of x that make the denominator equal to zero. Setting the denominator equal to zero, we solve:
9x + 4 = 0
9x = -4
x = -4/9
So, the domain of (f/g)(x) is (-∞, -4/9) U (-4/9, ∞).
To find (f+g)(x), we simply add the two functions f(x) and g(x):
(f+g)(x) = f(x) + g(x)
= 3x^2 - 9x + 9x + 4
= 3x^2 + 4
The domain for (f+g)(x) would be the same as the domain of f(x) and g(x), which is all real numbers.
Therefore, (f+g)(x) = 3x^2 + 4, and d = (-∞, ∞).
To find (f-g)(x), we subtract 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 for (f-g)(x) would be the same as the domain of f(x) and g(x), which is all real numbers.
Therefore, (f-g)(x) = 3x^2 - 18x - 4, and d = (-∞, ∞).
To find (fg)(x), we multiply 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 - 69x^2 - 36x
The domain for (fg)(x) would be the same as the domain of f(x) and g(x), which is all real numbers.
Therefore, (fg)(x) = 27x^3 - 69x^2 - 36x, and d = (-∞, ∞).
To find (f/g)(x), we divide f(x) by g(x):
(f/g)(x) = f(x) / g(x)
= (3x^2 - 9x) / (9x + 4)
However, we need to consider the restriction for division by zero. The denominator (9x + 4) cannot be equal to zero, so:
9x + 4 ≠ 0
9x ≠ -4
x ≠ -4/9
The domain for (f/g)(x) would be all real numbers except for x = -4/9.
Therefore, (f/g)(x) = (3x^2 - 9x) / (9x + 4), and d = (-∞, -4/9) U (-4/9, ∞).
To find the value of each expression and its respective domain, we need to perform the given operations on the functions f(x) and g(x) and determine the resulting function and its domain.
1. (f+g)(x):
To find (f+g)(x), we need to add f(x) and g(x) together:
(f+g)(x) = f(x) + g(x) = 3x^2 - 9x + (9x + 4) = 3x^2 - 9x + 9x + 4
Simplifying the expression, we get:
(f+g)(x) = 3x^2 + 4
The domain (d) for this function is all real numbers since there are no restrictions on the variable x.
Therefore, for (f+g)(x), the function is 3x^2 + 4, and the domain is (-∞, ∞).
2. (f-g)(x):
To find (f-g)(x), we need to subtract g(x) from f(x):
(f-g)(x) = f(x) - g(x) = 3x^2 - 9x - (9x + 4) = 3x^2 - 9x - 9x - 4
Simplifying the expression, we get:
(f-g)(x) = 3x^2 - 18x - 4
The domain (d) for this function is also all real numbers since there are no restrictions on the variable x.
Therefore, for (f-g)(x), the function is 3x^2 - 18x - 4, and the domain is (-∞, ∞).
3. (fg)(x):
To find (fg)(x), we need to multiply f(x) and g(x) together:
(fg)(x) = f(x) * g(x) = (3x^2 - 9x) * (9x + 4)
Using the distributive property and simplifying the expression, we get:
(fg)(x) = 27x^3 + 12x^2 - 81x^2 - 36x
Combining like terms, we have:
(fg)(x) = 27x^3 - 69x^2 - 36x
The domain (d) for this function is still all real numbers since there are no restrictions on the variable x.
Therefore, for (fg)(x), the function is 27x^3 - 69x^2 - 36x, and the domain is (-∞, ∞).
4. (f/g)(x):
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)
This expression cannot be simplified further, so we leave it as it is.
The domain (d) for this function is determined by the restriction that the denominator (g(x)) cannot be equal to zero to avoid division by zero:
9x + 4 ≠ 0
Solving for x, we have:
9x ≠ -4
x ≠ -4/9
Since x cannot be equal to -4/9, we exclude this value from the domain.
Therefore, the domain (d) for (f/g)(x) is (-∞, -4/9) U (-4/9, ∞).
In summary:
- (f+g)(x) = 3x^2 + 4, domain (d) = (-∞, ∞)
- (f-g)(x) = 3x^2 - 18x - 4, domain (d) = (-∞, ∞)
- (fg)(x) = 27x^3 - 69x^2 - 36x, domain (d) = (-∞, ∞)
- (f/g)(x) = (3x^2 - 9x) / (9x + 4), domain (d) = (-∞, -4/9) U (-4/9, ∞)