To determine the domain of the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex], we need to identify all the possible values of [tex]\( x \)[/tex] for which [tex]\( f(x) \)[/tex] is defined.
1. The given function is a polynomial of degree 2.
2. Polynomial functions, including quadratic functions like [tex]\( f(x) = x^2 + 3x + 5 \)[/tex], are defined for all real numbers.
3. There are no restrictions such as division by zero, taking the square root of a negative number, or logarithms which might limit the domain.
Therefore, the domain of the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex] is:
D. all real numbers
