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 results and domains of the given functions, follow these steps:
a) To find the product of f(x) and g(x), you need to multiply the two functions together: (f times g)(x) = f(x) * g(x).
So, (f times g)(x) = sqrt(x) * (3x - 2).
To determine the domain of (f times g)(x), you need to consider the domains of both f(x) and g(x) and determine their intersection.
1. The domain of f(x) = sqrt(x) is all non-negative real numbers (x ≥ 0).
2. The domain of g(x) = 3x - 2 is all real numbers since there are no restrictions on x.
Therefore, to find the domain of (f times g)(x), we need to consider the intersection of the domains of f(x) and g(x). Since both functions have no restrictions, the domain of (f times g)(x) is all real numbers.
b) To find the quotient of f(x) and g(x), you need to divide f(x) by g(x): (f/g)(x) = f(x) / g(x).
So, (f/g)(x) = sqrt(x) / (3x - 2).
To determine the domain of (f/g)(x), you need to consider the domains of f(x) and g(x) and any restrictions caused by division (i.e., avoiding division by zero).
1. The domain of f(x) = sqrt(x) is all non-negative real numbers (x ≥ 0).
2. The domain of g(x) = 3x - 2 is all real numbers since there are no restrictions on x, except when the denominator (3x - 2) equals zero.
To avoid division by zero, we solve for x where 3x - 2 = 0.
3x - 2 = 0
3x = 2
x = 2/3
Therefore, the only restriction is x ≠ 2/3.
Hence, the domain of (f/g)(x) is all non-negative real numbers excluding x = 2/3.