To determine the domain of the function [tex]\( f(x) = \frac{3}{x+2} - \sqrt{x-3} \)[/tex], we need to consider the constraints imposed by each part of the function.
1. Constraint from the square root term [tex]\(\sqrt{x-3}\)[/tex]:
- The expression inside the square root, [tex]\( x-3 \)[/tex], must be greater than or equal to zero for the square root to be defined in the real numbers. Therefore:
[tex]\[
x - 3 \geq 0
\][/tex]
[tex]\[
x \geq 3
\][/tex]
2. Constraint from the rational term [tex]\(\frac{3}{x+2}\)[/tex]:
- The denominator [tex]\( x+2 \)[/tex] must not be zero because division by zero is undefined. Therefore:
[tex]\[
x + 2 \neq 0
\][/tex]
[tex]\[
x \neq -2
\][/tex]
Since [tex]\( x \geq 3 \)[/tex] is a stronger constraint and automatically excludes [tex]\( x = -2 \)[/tex] (because [tex]\( -2 \)[/tex] is not greater than or equal to [tex]\( 3 \)[/tex]), we combine these constraints.
As a result, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[
x \geq 3
\][/tex]
Thus, the statement can be completed as:
"The domain for [tex]\( f(x) \)[/tex] is all real numbers greater than or equal to 3."
So, the filled blank should read "greater":
[tex]\[
\boxed{\text{greater}}
\][/tex]
