for the given functions f&g, find the following and state the domain of each result.
f(x)= 5x+4/7x-8; g(x)= 2x/7x-8
a. (f+g)(x)=?
what is the domain?
b. (f/g)(x)=?
what is the domain?
A). (7x+ 4)/(7x-8)
Domain is all except x can not equal 8/7 because it will make the denominator equal to 0
B). ( 5x + 4)/ 2x
Domain all except x can not equal 0
To find the results of (f+g)(x) and (f/g)(x), we need to add the two functions and divide one function by the other, respectively. Let's calculate them step by step:
a. (f+g)(x):
To find (f+g)(x), we need to add the functions f(x) and g(x) together.
(f+g)(x) = f(x) + g(x)
Given:
f(x) = (5x + 4)/(7x - 8)
g(x) = (2x)/(7x - 8)
Now, let's add f(x) and g(x):
(f+g)(x) = ((5x + 4)/(7x - 8)) + ((2x)/(7x - 8))
To simplify the expression, we need to find a common denominator, which is (7x - 8).
(f+g)(x) = (5x + 4 + 2x)/(7x - 8)
Combining the like terms in the numerator:
(f+g)(x) = (7x + 4)/(7x - 8)
The result is (f+g)(x) = (7x + 4)/(7x - 8)
Now, let's find the domain of (f+g)(x):
The domain of (f+g)(x) is the set of all values that x can take without causing any division by zero or any other undefined operations.
In this case, we must make sure that the denominator, 7x - 8, is not equal to zero. So, we solve the equation 7x - 8 ≠ 0 to find the domain:
7x - 8 ≠ 0
7x ≠ 8
x ≠ 8/7
Therefore, the domain of (f+g)(x) is all real numbers except x = 8/7.
b. (f/g)(x):
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) = (5x + 4)/(7x - 8)
g(x) = (2x)/(7x - 8)
Now, let's divide f(x) by g(x):
(f/g)(x) = ([(5x + 4)/(7x - 8)]) / ([(2x)/(7x - 8)])
To divide by a fraction, we multiply by its reciprocal:
(f/g)(x) = ([(5x + 4)/(7x - 8)]) * ([(7x - 8)/(2x)])
To further simplify, we can cancel out the common factor (7x - 8) in the numerator and denominator:
(f/g)(x) = (5x + 4) / (2x)
The result is (f/g)(x) = (5x + 4) / (2x)
Now, let's find the domain of (f/g)(x):
The domain of (f/g)(x) is the set of all values that x can take without causing any division by zero or any other undefined operations.
In this case, we must make sure that the denominator, 2x, is not equal to zero. So, we solve the equation 2x ≠ 0 to find the domain:
2x ≠ 0
x ≠ 0
Therefore, the domain of (f/g)(x) is all real numbers except x = 0.