To determine the domain of the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex], we need to consider the types of values that [tex]\( x \)[/tex] can take on such that the function [tex]\( f(x) \)[/tex] is well-defined.
1. Understanding polynomial functions: A polynomial function is one that can be represented in the form [tex]\( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)[/tex], where [tex]\( a_n, a_{n-1}, \ldots, a_1, a_0 \)[/tex] are constants, and [tex]\( n \)[/tex] is a non-negative integer. Polynomial functions are continuous and defined for all real numbers.
2. Analyzing the given function: The given function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex] is a quadratic polynomial. Quadratic polynomials are a specific type of polynomial of degree 2, and they are defined for all real numbers.
3. Checking for any restrictions: There are no restrictions such as division by zero or taking the square root of a negative number in this quadratic function. Hence, [tex]\( x \)[/tex] can take any real value.
4. Conclusion: Since there are no restrictions on [tex]\( x \)[/tex] for this quadratic function, the domain of the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex] is all real numbers.
Therefore, the correct answer is:
D. all real numbers
