Let f(x)=5x^3-2x and g(x)=3x^3.Preform the operation given and state the domain.
A. F(x)+ g(x)
B. F(x) - g(x)
C. F(x) • g(x)
D. F(x) / g(x)
E. f(g(x))
F. g (f(x)
That's "perform"
I'll assume you know how to add and multiply polynomials for A-D.
f(g) = 5g^3-2g = 5(3x^3)^3-2(3x^3) = 135x^9-6x^3
g(f) = 3f^3 = 3(5x^3-2x)^3 = 375x^9-450x^7+180x^5-24x^3
A. F(x) + g(x)
To perform this operation, we simply add the functions together.
F(x) + g(x) = (5x^3 - 2x) + (3x^3)
Combining like terms, we get:
F(x) + g(x) = 8x^3 - 2x
The domain for this sum is the same as the domains of the individual functions, which is all real numbers.
Domain: All real numbers
B. F(x) - g(x)
To perform this operation, we subtract g(x) from F(x).
F(x) - g(x) = (5x^3 - 2x) - (3x^3)
Combining like terms, we get:
F(x) - g(x) = 2x^3 - 2x
The domain for this difference is the same as the domains of the individual functions, which is all real numbers.
Domain: All real numbers
C. F(x) • g(x)
To perform this operation, we multiply the functions together.
F(x) • g(x) = (5x^3 - 2x) • (3x^3)
To simplify, we apply the distributive property:
F(x) • g(x) = 15x^6 - 6x^4
The domain for this product is the same as the domains of the individual functions, which is all real numbers.
Domain: All real numbers
D. F(x) / g(x)
To perform this operation, we divide F(x) by g(x).
F(x) / g(x) = (5x^3 - 2x) / (3x^3)
There is no common factor that can be cancelled out, so we leave the division as is.
Domain: All real numbers except x = 0 (since division by zero is undefined)
E. f(g(x))
To compute f(g(x)), we substitute g(x) into f(x).
f(g(x)) = 5(g(x))^3 - 2(g(x))
Since g(x) = 3x^3, we substitute it in:
f(g(x)) = 5(3x^3)^3 - 2(3x^3)
Performing the calculations, we simplify:
f(g(x)) = 5(27x^9) - 6x^3
f(g(x)) = 135x^9 - 6x^3
The domain for this composition is the same as the domain of g(x), which is all real numbers.
Domain: All real numbers
F. g(f(x))
To compute g(f(x)), we substitute f(x) into g(x).
g(f(x)) = 3(f(x))^3
Since f(x) = 5x^3 - 2x, we substitute it in:
g(f(x)) = 3((5x^3 - 2x))^3
Performing the calculations, we simplify:
g(f(x)) = 3(125x^9 - 150x^4 + 60x^2 - 8x)
g(f(x)) = 375x^9 - 450x^4 + 180x^2 - 24x
The domain for this composition is the same as the domain of f(x), which is all real numbers.
Domain: All real numbers
To perform the operations and determine the domain of each expression involving the functions f(x) and g(x), we will substitute the function expressions and then simplify.
Note: The domain refers to the set of all possible x-values for which the expression is defined.
Let's start with each operation:
A. F(x) + g(x):
We substitute the function expressions and add:
F(x) + g(x) = (5x^3 - 2x) + (3x^3)
Simplifying, we get:
F(x) + g(x) = 5x^3 - 2x + 3x^3
= 8x^3 - 2x
The domain for this expression is the set of all real numbers because there are no restrictions on x.
B. F(x) - g(x):
We substitute the function expressions and subtract:
F(x) - g(x) = (5x^3 - 2x) - (3x^3)
Simplifying, we get:
F(x) - g(x) = 5x^3 - 2x - 3x^3
= 2x^3 - 2x
The domain for this expression is also the set of all real numbers because there are no restrictions on x.
C. F(x) • g(x):
We substitute the function expressions and multiply:
F(x) • g(x) = (5x^3 - 2x) • (3x^3)
Simplifying, we get:
F(x) • g(x) = 15x^6 - 6x^4
The domain for this expression is the set of all real numbers because there are no restrictions on x.
D. F(x) / g(x):
We substitute the function expressions and divide:
F(x) / g(x) = (5x^3 - 2x) / (3x^3)
Simplifying, we get:
F(x) / g(x) = (5x^3 - 2x) / (3x^3)
= (5x^3 / 3x^3) - (2x / 3x^3)
= 5/3 - 2/x^2
The domain for this expression is all real numbers except x = 0 since division by zero is undefined.
E. f(g(x)):
We substitute the function expressions and then simplify:
f(g(x)) = f(3x^3) = 5(3x^3)^3 - 2(3x^3)
= 5(27x^9) - 6x^3
= 135x^9 - 6x^3
The domain for this expression is the set of all real numbers because there are no restrictions on x.
F. g(f(x)):
We substitute the function expressions and then simplify:
g(f(x)) = g(5x^3 - 2x) = 3(5x^3 - 2x)^3
= 3(125x^9 - 150x^7 + 60x^5 - 8x^3)
The domain for this expression is the set of all real numbers because there are no restrictions on x.
To summarize:
A. F(x) + g(x): Domain is all real numbers
B. F(x) - g(x): Domain is all real numbers
C. F(x) • g(x): Domain is all real numbers
D. F(x) / g(x): Domain is all real numbers except x = 0
E. f(g(x)): Domain is all real numbers
F. g(f(x)): Domain is all real numbers