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+2x-8 and g(x)=9x+5
(f+g)(x)=
d=
(f-g)(x)=
d=
(fg)(x)=
d=
(f/g)(x)=
d=
To find each of the expressions and their respective domains, we'll substitute the functions into the given expressions directly:
1. (f+g)(x) = f(x) + g(x) = (-8x^2+2x-8) + (9x+5) = -8x^2 + 11x - 3
The domain of (f+g)(x) is all real numbers, so its domain is (-∞, ∞).
2. (f-g)(x) = f(x) - g(x) = (-8x^2+2x-8) - (9x+5) = -8x^2 + 2x - 9x -13 = -8x^2 - 7x -13
The domain of (f-g)(x) is all real numbers, so its domain is (-∞, ∞).
3. (fg)(x) = f(x) * g(x) = (-8x^2+2x-8) * (9x+5) = -72x^3 - 32x^2 -40x - 40
The domain of (fg)(x) is still all real numbers, so its domain is (-∞, ∞).
4. (f/g)(x) = f(x) / g(x) = (-8x^2+2x-8) / (9x+5)
The domain of (f/g)(x) is all real numbers except for the values of x that make the denominator zero (since division by zero is undefined). So we need to solve the equation 9x+5=0 to find the excluded values:
9x = -5
x = -5/9
The excluded value is x = -5/9.
Therefore, the domain of (f/g)(x) is (-∞, -5/9) U (-5/9, ∞).
To find each of the given functions and their domains, we will perform the specified operations on f(x) and g(x).
1. (f+g)(x) = f(x) + g(x)
= (-8x^2+2x-8) + (9x+5)
= -8x^2 + (2x + 9x) + (-8 + 5)
= -8x^2 + 11x - 3
Domain of (f+g)(x) is all real numbers since there are no restrictions on x.
d = (-∞, +∞)
2. (f-g)(x) = f(x) - g(x)
= (-8x^2+2x-8) - (9x+5)
= -8x^2 + (2x - 9x) + (-8 - 5)
= -8x^2 - 7x - 13
Domain of (f-g)(x) is all real numbers since there are no restrictions on x.
d = (-∞, +∞)
3. (fg)(x) = f(x) * g(x)
= (-8x^2+2x-8) * (9x+5)
= (-8x^2)*(9x) + (-8x^2)*(5) + (2x)*(9x) + (2x)*(5) + (-8)*(9x) + (-8)*(5)
= -72x^3 - 40x^2 + 18x^2 + 10x - 72x - 40
Domain of (fg)(x) is all real numbers since there are no restrictions on x.
d = (-∞, +∞)
4. (f/g)(x) = f(x) / g(x)
= (-8x^2+2x-8) / (9x+5)
The domain of (f/g)(x) includes all values of x where the denominator (9x+5) is not equal to zero. Solving for (9x+5) ≠ 0:
9x + 5 ≠ 0
9x ≠ -5
x ≠ -5/9
Therefore, the domain of (f/g)(x) is all real numbers except x = -5/9.
d = (-∞, -5/9) U (-5/9, +∞)
To find each of the following functions and their respective domains, we will perform the given operations on the given functions f(x) and g(x). Let's analyze each case step by step:
1. (f+g)(x):
To find (f+g)(x), we simply add the two functions, f(x) and g(x), together. The sum of two functions is calculated by adding their corresponding terms.
(f+g)(x) = f(x) + g(x) = (-8x^2+2x-8) + (9x+5)
= -8x^2 + 2x - 8 + 9x + 5
= -8x^2 + 11x - 3
The resulting function is -8x^2 + 11x - 3. To determine the domain of this function, we need to consider any restrictions on x that would make the function undefined.
In this case, there are no restrictions on x for polynomial functions like this one. Therefore, the domain of (f+g)(x) is all real numbers. We can express this using interval notation as (-∞, ∞).
2. (f-g)(x):
To find (f-g)(x), we subtract the function g(x) from f(x), again by subtracting their corresponding terms.
(f-g)(x) = f(x) - g(x) = (-8x^2+2x-8) - (9x+5)
= -8x^2 + 2x - 8 - 9x - 5
= -8x^2 - 7x - 13
The resulting function is -8x^2 - 7x - 13. To determine the domain, again, there are no restrictions on x for this polynomial function. Therefore, the domain of (f-g)(x) is all real numbers (-∞, ∞).
3. (fg)(x):
To find (fg)(x), we multiply the two functions, f(x) and g(x), together. We distribute each term of f(x) across each term of g(x).
(fg)(x) = f(x) * g(x) = (-8x^2+2x-8) * (9x+5)
= -8x^2 * (9x+5) + 2x * (9x+5) - 8 * (9x+5)
= -72x^3 - 40x^2 + 18x^2 + 10x - 72x - 40 + 40
= -72x^3 - 22x^2 - 62x
The resulting function is -72x^3 - 22x^2 - 62x. The domain of this function is again all real numbers (-∞, ∞).
4. (f/g)(x):
To find (f/g)(x), we divide the function f(x) by g(x) by dividing each term of f(x) by each term of g(x).
(f/g)(x) = f(x) / g(x) = (-8x^2+2x-8) / (9x+5)
It is important to note that for division, we need to consider any values of x that make the denominator equal to zero, as division by zero is undefined.
To find the values of x that would make (f/g)(x) undefined, we set the denominator equal to zero and solve for x:
9x + 5 = 0
9x = -5
x = -5/9
So, x = -5/9 is a restriction for (f/g)(x). To find the domain, we exclude this value from the domain of all real numbers. Therefore, the domain of (f/g)(x) is all real numbers except x = -5/9. Using interval notation, we can express this as (-∞, -5/9) U (-5/9, ∞).
To summarize:
1. (f+g)(x) = -8x^2 + 11x - 3; domain: (-∞, ∞)
2. (f-g)(x) = -8x^2 - 7x - 13; domain: (-∞, ∞)
3. (fg)(x) = -72x^3 - 22x^2 - 62x; domain: (-∞, ∞)
4. (f/g)(x) = (-8x^2+2x-8) / (9x+5); domain: (-∞, -5/9) U (-5/9, ∞)