For the given functions f and g, find the following and state the domain of each result
f(x)=sqrtx;g(x)=3x-2
a) (f times g)(x)=___
What is the domain of f times g?
b) (f/g)(x)=____
what is the domain of f/g?
To find the result of the given functions and determine the domain, we need to follow these steps:
a) Find (f times g)(x):
To find (f times g)(x), we multiply the two functions f(x) and g(x):
(f times g)(x) = f(x) * g(x)
Substituting f(x) and g(x):
(f times g)(x) = sqrt(x) * (3x-2)
Now, simplify the expression:
(f times g)(x) = 3x*sqrt(x) - 2*sqrt(x)
The result of (f times g)(x) is 3x*sqrt(x) - 2*sqrt(x).
To determine the domain of (f times g)(x), we need to consider the domains of both functions involved.
The function f(x) = sqrt(x) is defined for non-negative values of x since the square root of a negative number is undefined in the real number system. So, the domain of f(x) is x ≥ 0.
The function g(x) = 3x - 2 is a linear function defined for all real values of x. So, there are no restrictions on the domain of g(x).
Since the domain of (f times g)(x) is determined by the common domain of f(x) and g(x), the domain of (f times g)(x) is x ≥ 0.
b) Find (f/g)(x):
To find (f/g)(x), we divide f(x) by g(x):
(f/g)(x) = f(x) / g(x)
Substituting f(x) and g(x):
(f/g)(x) = sqrt(x) / (3x-2)
Now, consider the domain of (f/g)(x).
The function f(x) = sqrt(x) is defined for x ≥ 0.
The function g(x) = 3x - 2 can be any real value except for x = 2/3 to avoid division by zero.
Since the denominator in (f/g)(x) cannot be zero, we need to exclude the value x = 2/3 from the domain.
Thus, the domain of (f/g)(x) is x ≥ 0, excluding x = 2/3.