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?
Please show work
To find (f+g)(x), we need to add the two functions f(x) and g(x) together.
a. (f+g)(x) = f(x) + g(x)
Let's substitute the given functions:
(f+g)(x) = (5x+4)/(7x-8) + (2x)/(7x-8)
To add these two fractions, we need a common denominator. In this case, the denominator is already the same for both fractions, which is (7x-8). Therefore, we can add the numerators:
(f+g)(x) = (5x+4+2x)/(7x-8)
Simplifying further, we have:
(f+g)(x) = (7x+4)/(7x-8)
The domain of a function is the set of all possible values of x for which the function is defined. In this case, the only restriction we have is that the denominator (7x-8) should not equal zero since division by zero is undefined. Thus, we set the denominator equal to zero and solve for x:
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) = f(x) / g(x)
Similarly, let's substitute the given functions:
(f/g)(x) = (5x+4)/(7x-8) / (2x)/(7x-8)
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
(f/g)(x) = (5x+4)/(7x-8) * (7x-8)/(2x)
Simplifying further, we have:
(f/g)(x) = (5x+4) / (2x)
Again, the domain of a function is the set of all possible values of x for which the function is defined. In this case, the only restriction we have is that the denominator (2x) should not equal zero since division by zero is undefined. Thus, we set the denominator equal to zero and solve for x:
2x ≠ 0
x ≠ 0
Therefore, the domain of (f/g)(x) is all real numbers except x = 0.