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)=-8x^2+x-8 and g(x)=9x+5
(f+g)(x)=
d=
(f-g)(x)=
d=
(fg)(x)=
d=
(f/g)(x)=
d=
To find (f+g)(x), we simply add the two functions together:
(f+g)(x) = f(x) + g(x) = (-8x^2 + x - 8) + (9x + 5) = -8x^2 + 10x - 3
The domain of (f+g)(x) is the set of all real numbers since there are no restrictions on the domain of addition. So the domain is (-∞, ∞).
To find (f-g)(x), we subtract g(x) from f(x):
(f-g)(x) = f(x) - g(x) = (-8x^2 + x - 8) - (9x + 5) = -8x^2 + x - 9x - 13 = -8x^2 - 8x - 13
The domain of (f-g)(x) is also the set of all real numbers. So the domain is (-∞, ∞).
To find (f*g)(x), we multiply f(x) and g(x):
(f*g)(x) = f(x)*g(x) = (-8x^2 + x - 8) * (9x + 5) = -72x^3 - 20x^2 + 9x^2 + 63x + 45x + 40 = -72x^3 - 11x^2 + 108x + 40
The domain of (f*g)(x) is again the set of all real numbers. So the domain is (-∞, ∞).
To find (f/g)(x), we divide f(x) by g(x):
(f/g)(x) = f(x)/g(x) = (-8x^2 + x - 8) / (9x + 5)
The domain of (f/g)(x) is the set of all real numbers except for the values of x that make the denominator equal to zero, since division by zero is undefined. So we need to find the values of x that make (9x + 5) = 0:
9x + 5 = 0
9x = -5
x = -5/9
Therefore, x = -5/9 is not in the domain of (f/g)(x).
The domain of (f/g)(x) is (-∞, -5/9) U (-5/9, ∞).
To find each of the given functions and their respective domains, we will perform the operations step by step:
1. (f+g)(x):
To find (f+g)(x), we add the given functions f(x) and g(x):
(f+g)(x) = f(x) + g(x) = (-8x^2+x-8) + (9x+5)
(f+g)(x) = -8x^2 + 10x - 3
2. Domain of (f+g)(x):
The domain is the set of all possible input values for which the function is defined. Since we are not given any restrictions or limitations, the function (f+g)(x) is defined for all real numbers. Therefore, the domain is (-∞, ∞) using interval notation.
3. (f-g)(x):
To find (f-g)(x), we subtract the given functions f(x) and g(x):
(f-g)(x) = f(x) - g(x) = (-8x^2+x-8) - (9x+5)
(f-g)(x) = -8x^2 + x - 9x - 13
(f-g)(x) = -8x^2 - 8x - 13
4. Domain of (f-g)(x):
Similar to above, the domain of (f-g)(x) is (-∞, ∞).
5. (fg)(x):
To find (fg)(x), we multiply the given functions f(x) and g(x):
(fg)(x) = f(x) * g(x) = (-8x^2+x-8) * (9x+5)
(fg)(x) = -72x^3 - 40x^2 + 9x^2 + 5x - 72x - 40
(fg)(x) = -72x^3 - 31x^2 - 67x - 40
6. Domain of (fg)(x):
Again, the domain of (fg)(x) is (-∞, ∞).
7. (f/g)(x):
To find (f/g)(x), we divide the given functions f(x) by g(x):
(f/g)(x) = f(x) / g(x) = (-8x^2+x-8) / (9x+5)
7a. Solving for the domain of (f/g)(x):
For division, we need to ensure that the denominator (9x+5) is not equal to zero. So, we solve the equation: 9x+5=0
9x = -5
x = -5/9
Therefore, the function (f/g)(x) is defined for all real numbers except x = -5/9. The domain is (-∞, -5/9) U (-5/9, ∞). This is the domain of (f/g)(x).
To summarize:
- (f+g)(x) = -8x^2 + 10x - 3
Domain: (-∞, ∞)
- (f-g)(x) = -8x^2 - 8x - 13
Domain: (-∞, ∞)
- (fg)(x) = -72x^3 - 31x^2 - 67x - 40
Domain: (-∞, ∞)
- (f/g)(x) = (-8x^2+x-8) / (9x+5)
Domain: (-∞, -5/9) U (-5/9, ∞)
To find each of the required functions and their respective domains, we'll need to perform the given operations on the functions f(x) and g(x) and determine the resulting function and its domain for each case.
1. (f+g)(x) - This is the sum of the two functions f(x) and g(x). To find it, simply add the functions together by combining like terms:
(f+g)(x) = f(x) + g(x) = (-8x^2+x-8) + (9x+5)
After combining like terms, we get:
(f+g)(x) = -8x^2 + 10x - 3
Now, let's determine the domain for (f+g)(x):
The domain of a sum of functions is the intersection of the domains of the individual functions. Both f(x) and g(x) are defined for all real numbers, so their intersection is also all real numbers.
Therefore, the domain of (f+g)(x) is (-∞, ∞) (all real numbers).
2. (f-g)(x) - This is the difference between the functions f(x) and g(x). To find it, subtract the second function from the first:
(f-g)(x) = f(x) - g(x) = (-8x^2+x-8) - (9x+5)
After subtracting, we get:
(f-g)(x) = -8x^2 - 8x - 13
For the domain of (f-g)(x), we again look at the intersection of the domains of f(x) and g(x). As mentioned before, both f(x) and g(x) are defined for all real numbers, so their intersection is (-∞, ∞).
Therefore, the domain of (f-g)(x) is also (-∞, ∞).
3. (fg)(x) - This represents the product of the two functions f(x) and g(x). To find it, multiply the two functions together:
(fg)(x) = f(x) * g(x) = (-8x^2+x-8) * (9x+5)
Multiplying the functions gives:
(fg)(x) = -72x^3 - 37x^2 - 45x - 40
Let's determine the domain for (fg)(x):
Similar to before, the domain of a product of functions is the intersection of the individual function's domains. Both f(x) and g(x) are defined for all real numbers, so their intersection is (-∞, ∞).
Therefore, the domain of (fg)(x) is (-∞, ∞).
4. (f/g)(x) - This represents the quotient of the two functions f(x) and g(x). To find it, divide f(x) by g(x):
(f/g)(x) = f(x) / g(x) = (-8x^2+x-8) / (9x+5)
The quotient cannot be simplified any further.
To find the domain of (f/g)(x), we need to consider the restrictions of division. Division by zero is not defined, so we need to find any values of x for which the denominator is equal to zero. In this case, we solve the equation:
9x+5 = 0
9x = -5
x = -5/9
Therefore, the domain of (f/g)(x) is all real numbers except x = -5/9. We can represent this in interval notation as (-∞, -5/9) U (-5/9, ∞).