for the given functions f and g find the following and state the domain of each result
f(x)=3x+1/8x-9; g(x)=5x/8x-9
A) (f+g)(x)=?
B)domain of (f+g)
c) (f/g)(x)=?
d) domain of f/g?
please show work
To find (f+g)(x), we need to add the two functions f(x) and g(x).
A) (f+g)(x) = f(x) + g(x)
Given f(x) = (3x+1)/(8x-9) and g(x) = 5x/(8x-9), we can substitute these values in:
(f+g)(x) = (3x+1)/(8x-9) + 5x/(8x-9)
Next, we need to find a common denominator, which is already (8x-9) in this case. So, we can combine the numerators:
(f+g)(x) = [(3x+1) + 5x] / (8x-9)
(f+g)(x) = (8x+1) / (8x-9)
B) To find the domain of (f+g)(x), we need to consider any restrictions on the variable x that would make the denominator zero, as division by zero is undefined.
In this case, the denominator (8x-9) should not be equal to zero:
8x - 9 ≠ 0
Solving the equation, we find:
8x ≠ 9
x ≠ 9/8
Therefore, the domain of (f+g)(x) is all real numbers except x = 9/8.
C) To find (f/g)(x), we need to divide the function f(x) by g(x).
(f/g)(x) = f(x) / g(x)
Given f(x) = (3x+1)/(8x-9) and g(x) = 5x/(8x-9), we can substitute these values in:
(f/g)(x) = [(3x+1)/(8x-9)] / [(5x)/(8x-9)]
Next, we multiply the numerator by the reciprocal of the denominator:
(f/g)(x) = [(3x+1)/(8x-9)] * [(8x-9)/(5x)]
(f/g)(x) = (3x+1)/(5x)
D) To find the domain of (f/g)(x), we need to consider any restrictions on the variable x that would make the denominator zero.
In this case, the denominator (5x) should not be equal to zero:
5x ≠ 0
Solving the equation, we find:
x ≠ 0
Therefore, the domain of (f/g)(x) is all real numbers except x = 0.