Answer:
Let's find each function step by step:
a. (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)
Given that f(x) = x^2 + 2x and g(x) = 5 - x, we can substitute these into the equation:
(f+g)(x) = (x^2 + 2x) + (5 - x)
Now, we simplify by combining like terms:
(f+g)(x) = x^2 + 2x + 5 - x
(f+g)(x) = x^2 + x + 5
So, (f+g)(x) = x^2 + x + 5.
The domain of (f+g)(x) is the set of all real numbers since there are no restrictions on the domain for polynomial functions.
b.(g-f)(x)
To find (g-f)(x), we need to subtract the function f(x) from the function g(x):
(g-f)(x) = g(x) - f(x)
Given that f(x) = x^2 + 2x and g(x) = 5 - x, we can substitute these into the equation:
(g-f)(x) = (5 - x) - (x^2 + 2x)
Now, we simplify by combining like terms:
(g-f)(x) = 5 - x - x^2 - 2x
(g-f)(x) = -x^2 - 3x + 5
So, (g-f)(x) = -x^2 - 3x + 5.
The domain of \((g-f)(x)\) is the set of all real numbers since there are no restrictions on the domain for polynomial functions.
c. (f⋅g)(x)
To find (f⋅g)(x), we need to multiply the functions f(x) and g(x):
(f⋅g)(x) = f(x) \cdot g(x)
Given that f(x) = x^2 + 2x and g(x) = 5 - x, we can substitute these into the equation:
(f⋅g)(x) = (x^2 + 2x) \cdot (5 - x)
Now, we simplify by using the distributive property:
(f⋅g)(x) = x^2(5 - x) + 2x(5 - x)
(f⋅g)(x) = 5x^2 - x^3 + 10x - 2x^2
(f⋅g)(x) = -x^3 + 3x^2 + 10x
So, (f⋅g)(x) = -x^3 + 3x^2 + 10x.
The domain of (f⋅g)(x) is the set of all real numbers since there are no restrictions on the domain for polynomial functions.
d. (g/f)(x)
To find (g/f)(x), we need to divide the function g(x) by the function f(x):
(g/f)(x) = \frac{g(x)}{f(x)}
Given that f(x) = x^2 + 2x and g(x) = 5 - x, we can substitute these into the equation:
(g/f)(x) = \frac{5 - x}{x^2 + 2x}
The domain of (g/f)(x) consists of all real numbers except for the values of x that make the denominator x^2 + 2x equal to zero, as division by zero is undefined. To find these values, we solve the equation x^2 + 2x = 0:
x(x + 2) = 0
x = 0 or x + 2 = 0
x = 0 or x = -2
Therefore, the domain of (g/f)(x) is all real numbers except x = 0 and x = -2.
