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
(they look like fractions)
A) (f+g)(x)=?
B)domain of (f+g)
c) (f/g)(x)=?
d) domain of f/g?
please show work
a. (f+g)x = (3x+1)/(8x-9) + 5x/(8x-9) =
(8x+1)/(8x-9).
b. 8x-9 = 0
8x = 9
X = 9/8.
When X = 9/8, the denominator becomes zero and the fraction undefined. Therefore, 9/8 is out of the domain.
Domain = All real values of X except 9/8: 9/8 > X > 9/8.
c. F(x)/G(x) = F(x) * 1/G(x).
F(x)/G(x) = (3x+1)/(8x-9) * (8x-9)/5x=
(3x+1)/5x.
d. The domain = all real values of X:
-Infinity < X < +Infinity.
A) To find (f+g)(x), we need to add the two functions f(x) and g(x):
(f+g)(x) = f(x) + g(x)
= (3x+1)/(8x-9) + (5x)/(8x-9)
We cannot simplify this any further.
B) The domain of (f+g)(x) will be the set of all real numbers except for the values of x that make the denominator 0. In this case, the denominator is 8x-9. So, 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))
= (3x+1)/(8x-9) / (5x)/(8x-9)
Dividing by a fraction is the same as multiplying by the reciprocal, so we can rewrite this as:
(f/g)(x) = (3x+1)/(8x-9) * (8x-9)/(5x)
Simplifying,
(f/g)(x) = (3x+1)/(5x)
D) The domain of (f/g)(x) will be the set of all real numbers except for the values of x that make the denominator 0. In this case, the denominator is 5x. So, the domain of (f/g)(x) is all real numbers except x = 0.
To find the answers for each part of the question, we will perform the corresponding operations on the given functions f(x) and g(x). Let's go step by step:
Given functions:
f(x) = (3x + 1) / (8x - 9)
g(x) = (5x) / (8x - 9)
A) To find (f + g)(x), we need to add the functions f(x) and g(x).
(f + g)(x) = f(x) + g(x)
= (3x + 1) / (8x - 9) + (5x) / (8x - 9)
= (3x + 5x + 1) / (8x - 9)
= (8x + 1) / (8x - 9)
B) The domain of (f + g) is the set of values for which the expression (8x - 9) in the denominator is not equal to zero. When the denominator is zero, the expression is undefined.
So, we need to solve the equation:
8x - 9 ≠ 0
8x ≠ 9
x ≠ 9/8
Therefore, the domain of (f + g) is all real numbers except x = 9/8.
C) To find (f/g)(x), we have to divide the function f(x) by g(x).
(f/g)(x) = f(x) / g(x)
= ((3x + 1) / (8x - 9)) / ((5x) / (8x - 9))
= (3x + 1) / (5x)
D) The domain of (f/g) is the set of values for which the expression (5x) in the denominator is not equal to zero.
So, we need to solve the equation:
5x ≠ 0
x ≠ 0
Therefore, the domain of (f/g) is all real numbers except x = 0.
To summarize:
A) (f + g)(x) = (8x + 1) / (8x - 9)
B) The domain of (f + g) is all real numbers except x = 9/8.
C) (f/g)(x) = (3x + 1) / (5x)
D) The domain of (f/g) is all real numbers except x = 0.