For the given functions f and g find the following and state the domain of each result f(x)=5x+1/4x-7; g(x) = 3x/4x-7
find the following what?
Domain of f and g is all reals except x=7/4
a) (f+g)(x)=
what is the domain of f+g?
b) f/g(x)=
What is the domain of f/g?
since f and g have the same domain, f+g has the same domain as f and g.
f/g = (5x+1)/(3x)
Now, we have to exclude x=7/4 because neither f nor g is defined there.
In addition, we have to exclude x=0 because f/g is not defined there.
So, domain of f/g is all reals except 0,7/4
f+g(x)=?
eh? Just add 'em up:
(f+g)(x)
= f(x)+g(x)
= (5x+1)/(4x-7) + 3x/(4x-7)
= (5x+1+3x)/(4x-7)
= (8x+1)/(4x-7)
To find the results, we need to perform various operations on the given functions, f(x) and g(x). Let's calculate each part step by step:
1. Finding f(x) + g(x):
To find the sum of two functions, we need to add the functions' values at the same input. Substituting f(x) and g(x) into the equation:
f(x) + g(x) = (5x + 1)/(4x - 7) + (3x)/(4x - 7)
2. Finding f(x) - g(x):
To find the difference of two functions, we need to subtract the functions' values at the same input. Substituting f(x) and g(x) into the equation:
f(x) - g(x) = (5x + 1)/(4x - 7) - (3x)/(4x - 7)
3. Finding f(x) * g(x):
To find the product of two functions, we need to multiply the functions' values at the same input. Substituting f(x) and g(x) into the equation:
f(x) * g(x) = [(5x + 1)/(4x - 7)] * [(3x)/(4x - 7)]
4. Finding f(x) / g(x):
To find the quotient of two functions, we need to divide the functions' values at the same input. Substituting f(x) and g(x) into the equation:
f(x) / g(x) = [(5x + 1)/(4x - 7)] / [(3x)/(4x - 7)]
Now let's determine the domain for each result:
- For f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x) / g(x):
The domain will be any real number (x), except for the values that make the denominator (4x - 7) equal to zero. So, the domain is all real numbers except x = 7/4.
To summarize:
- Domain for f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x) / g(x): All real numbers except x = 7/4.