For the given functions f and g find the following and state the domain of each.
f(x)=sqrt x; g(x)=7x-3
A.what is the domain of f*g?
B. What is the domain of f/g?
Please show all work.
domain of √x is x>=0
domain of 7x-3 is all reals
f*g = √x (7x-3) domain is now restricted to x>=0
f/g = √x/(7x-3) now the domain excludes values where the denominator is zero.
So, domain is all x>=0 except x = 3/7
To find the domain of the composite functions f*g and f/g,
1. First, we need to understand the domains of the individual functions f(x) = √x and g(x) = 7x - 3.
The domain of f(x) = √x is the set of all real numbers x such that x ≥ 0. This is because the square root function (√) is defined only for non-negative (or zero) values of x.
The domain of g(x) = 7x - 3 is the set of all real numbers since there are no restrictions on the input x.
Now, let's find the domains of the composite functions:
A. f*g:
To find the domain of f*g, we need to consider the composition of f(x) and g(x): (f*g)(x) = f(g(x)).
Substituting g(x) into f(x), we get:
(f*g)(x) = f(g(x)) = f(7x - 3) = √(7x - 3).
Since the square root (√) function is defined only for non-negative values, we need to find the values of x for which 7x - 3 ≥ 0.
7x - 3 ≥ 0
7x ≥ 3
x ≥ 3/7
Therefore, the domain of f*g is the set of all real numbers x such that x ≥ 3/7.
B. f/g:
To find the domain of f/g, we need to consider the division of f(x) by g(x): (f/g)(x) = f(x) / g(x).
Substituting f(x) and g(x), we get:
(f/g)(x) = f(x) / g(x) = (√x) / (7x - 3).
To determine the domain, we need to find the values of x for which the denominator, (7x - 3), is not equal to zero. We cannot divide by zero.
Solving 7x - 3 ≠ 0:
7x ≠ 3
x ≠ 3/7
Therefore, the domain of f/g is the set of all real numbers x excluding x = 3/7.
In summary:
A. The domain of f*g is x ≥ 3/7.
B. The domain of f/g is all real numbers excluding x = 3/7.