To determine the domain of the function [tex]\( f(x) = 2x^2 + 5 \sqrt{x - 2} \)[/tex], we should look at the conditions under which the function is defined. In this function, there are two main components to consider:
1. The term [tex]\( 2x^2 \)[/tex], which is a polynomial. Since polynomials are defined for all real numbers, there are no restrictions from this term.
2. The term [tex]\( 5 \sqrt{x - 2} \)[/tex], which involves a square root. For the square root to be defined, the expression inside the square root, [tex]\( x - 2 \)[/tex], must be non-negative. This means [tex]\( x - 2 \geq 0 \)[/tex].
Solving for [tex]\( x \)[/tex]:
[tex]\[ x - 2 \geq 0 \][/tex]
[tex]\[ x \geq 2 \][/tex]
Therefore, the restriction on [tex]\( x \)[/tex] comes from the requirement that [tex]\( x \geq 2 \)[/tex].
Combining these considerations, the domain of the function [tex]\( f(x) = 2x^2 + 5 \sqrt{x - 2} \)[/tex] is all real numbers greater than or equal to 2.
So, the complete statement is:
The domain for [tex]\( f(x) \)[/tex] is all real numbers greater than or equal to 2.
