For the given functions f & g find the following and state the domain of each result.
f(x)square root x;g(x)=5x-1
a.(f*g)(x)=
What is the domain of f*g?
b. (f/g)(x)=
What is the domain of f/g?
To find the function (f*g)(x), we need to multiply the two functions f(x) and g(x).
a. (f*g)(x) = f(x) * g(x) = (square root x) * (5x-1)
To simplify this expression, we multiply the terms:
(f*g)(x) = (square root x) * (5x-1)
= 5x * square root x - square root x
Now let's determine the domain of (f*g)(x). The domain represents the set of all possible values that x can take.
For the function (f*g)(x) = (square root x) * (5x-1) to be defined, both square root x and (5x-1) must be defined.
The square root function is defined for non-negative real numbers, so the first condition is square root x ≥ 0, which implies x ≥ 0.
The second condition is that the expression 5x - 1 should not result in dividing by zero. Thus, we set the denominator of 5x - 1 to be not equal to zero:
5x - 1 ≠ 0
5x ≠ 1
x ≠ 1/5
Combining these conditions, we find the domain of (f*g)(x) is x ≥ 0 and x ≠ 1/5.
b. To find the function (f/g)(x), we need to divide the function f(x) by g(x):
(f/g)(x) = f(x) / g(x) = (square root x) / (5x-1)
To simplify this expression, we divide the terms:
(f/g)(x) = (square root x) / (5x-1)
Let's now determine the domain of (f/g)(x).
For (f/g)(x) to be defined, the denominator 5x-1 must not be equal to zero:
5x - 1 ≠ 0
5x ≠ 1
x ≠ 1/5
So, the domain of (f/g)(x) is x ≠ 1/5.