For the given functions f and g, find the following and state the domain of each result.
f(x)=√x;g(x)=7x-2
(a)(f*g)(x)=
What is the domain of f*g?
(b) (f/g)(x)=
What is the domain of f/g?
To find (f*g)(x), we need to multiply the two functions f(x) and g(x) together.
(a) (f*g)(x) = f(x) * g(x) = (√x) * (7x-2)
To simplify this expression, we need to distribute the square root to both terms inside the parentheses:
(f*g)(x) = √x * 7x - √x * 2
Now we can simplify each term:
(f*g)(x) = 7x^(3/2) - 2√x
The domain of (f*g)(x) is the set of all values of x for which the expression is defined. Since both functions inside the expression are defined for all real numbers, the domain of (f*g)(x) is also all real numbers.
(b) (f/g)(x) = f(x) / g(x) = (√x) / (7x-2)
To find the domain of (f/g)(x), we need to determine the values of x for which the denominator is not equal to zero. In this case, we want to find the values of x that make (7x-2) ≠ 0.
Solving the equation (7x-2) = 0, we find:
7x = 2
x = 2/7
So, (f/g)(x) is defined for all real numbers except x = 2/7. Therefore, the domain of (f/g)(x) is all real numbers except x = 2/7.