For the given functions f and g, find the following and state the domain of each result.
f(x)=sqrtx;g(x)=5x-9
(a) (f*g)(x)=
What is the domain of f*g?
(b) f/g(x)=
a. (F*g)(x) = sqrt(x)(5x-9)
Domain: X => 0.
Domain: All real values of X that are equal to or greater than zero. The value
under the radical cannot be less than zero.
b. F/g(x) = sqrt(x)/(5x-9)
To find the functions (f * g)(x) and f/g(x), let's go step by step:
(a) To find (f * g)(x), we need to multiply the functions f(x) and g(x). We can write it as:
(f * g)(x) = f(x) * g(x)
Substituting the given functions:
(f * g)(x) = sqrt(x) * (5x - 9)
Simplifying further:
(f * g)(x) = 5x * sqrt(x) - 9 * sqrt(x)
So, (f * g)(x) = 5x * sqrt(x) - 9 * sqrt(x)
To determine the domain of (f * g)(x), we need to consider the domains of the individual functions involved.
For f(x) = sqrt(x), the domain is x ≥ 0 since square root is only defined for non-negative numbers.
For g(x) = 5x - 9, there are no restrictions on the domain as it is defined for all real numbers.
Combining these restrictions, the domain of (f * g)(x) is x ≥ 0.
(b) To find f/g(x), we need to divide the function f(x) by the function g(x). We can write it as:
f/g(x) = f(x) / g(x)
Substituting the given functions:
f/g(x) = sqrt(x) / (5x - 9)
So, f/g(x) = sqrt(x) / (5x - 9)
To determine the domain of f/g(x), we need to consider the domain restrictions of the individual functions involved.
For f(x) = sqrt(x), the domain is x ≥ 0.
For g(x) = 5x - 9, there are no restrictions on the domain as it is defined for all real numbers.
However, we must exclude values of x that make the denominator zero since division by zero is undefined.
So, 5x - 9 ≠ 0
Solving for x:
5x ≠ 9
x ≠ 9/5
Therefore, the domain of f/g(x) is x ≥ 0, excluding x = 9/5.
To find (f*g)(x), we need to multiply the two given functions f(x) and g(x) together and evaluate the result.
(a) (f*g)(x) = f(x) * g(x)
= (sqrt(x)) * (5x - 9)
Now, we can simplify this expression:
= (5x^3/2 - 9x^1/2)
To determine the domain of (f*g)(x), we need to consider any restrictions on x that would make the expression undefined. In this case, since the only operation involved is multiplication, there are no restrictions on the domain. Therefore, the domain of (f*g)(x) is all real numbers.
(b) To find f/g(x), we need to divide the function f(x) by the function g(x).
f/g(x) = f(x) / g(x)
= sqrt(x) / (5x - 9)
We need to consider any restrictions on x that would make the expression undefined. In this case, since dividing by zero is undefined, we need to find the values of x that would make the denominator equal to zero.
So, setting the denominator equal to zero, we solve:
5x - 9 = 0
5x = 9
x = 9/5
Therefore, x = 9/5 is the only value that makes the expression undefined. So, the domain of f/g(x) is all real numbers except x = 9/5.