F(x)=sqrt(9-3x) determine the domain of f

How would I break that out to determine f

I assume you mean the domain of x.

x has to be real, and has to be such that
9-3x>=0 or
x<=3

you are finding √(9-3x)

For this to result in a real number, 9-3x cannot be a negative number

so the domain is
9-3x ≥ 0
-3x ≥ -9
x ≤ 3

To determine the domain of the function f(x) = √(9-3x), we need to consider the values of x for which the function is defined.

In this case, the function f(x) is defined for any value of x that makes the expression inside the square root (√) nonnegative.

The expression inside the square root is 9-3x. For this expression to be nonnegative, it must be greater than or equal to zero.

Therefore, we can set up the inequality 9-3x ≥ 0 and solve it to find the values of x that satisfy this condition.

Here are the steps to solve the inequality:

1. Subtract 9 from both sides of the inequality: -3x ≥ -9.

2. Divide both sides of the inequality by -3. When dividing by a negative number, the inequality sign should be flipped, so the inequality becomes x ≤ 3.

So, the domain of the function f(x) = √(9-3x) is all values of x less than or equal to 3.

In interval notation, we can represent the domain as (-∞, 3].

This means that any value of x that is less than or equal to 3, including 3 itself, is in the domain of f(x).