To find the domain of the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex], we need to determine the set of all possible values of [tex]\( x \)[/tex] for which the function is defined.
1. Identify the type of function:
The function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex] is a polynomial function. Polynomial functions are expressions that involve variables raised to positive integer powers, combined using addition, subtraction, and multiplication.
2. Properties of polynomial functions:
Polynomial functions are defined for all real numbers because they do not have any restrictions such as division by zero or taking the square root of a negative number.
3. Conclude the domain:
Since polynomial functions are defined for all real numbers, the domain of [tex]\( f(x) = x^2 + 3x + 5 \)[/tex] includes every real number.
Thus, the domain of the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex] is:
D. all real numbers
