To determine the domain of the function [tex]\( f(x) = x^2 + x - 12 \)[/tex], we need to consider the type of function it is. The function [tex]\( f(x) \)[/tex] is a quadratic function.
A quadratic function is generally of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants. The function [tex]\( f(x) = x^2 + x - 12 \)[/tex] fits into this form, where [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -12 \)[/tex].
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Key points regarding quadratic functions:
1. Quadratic functions are polynomials of degree 2.
2. Polynomials are defined for all real numbers because there are no restrictions on the values [tex]\( x \)[/tex] can take (there are no divisions by zero, nor square roots of negative numbers to consider).
Therefore, the domain of a quadratic function is all real numbers, which can be expressed in interval notation as [tex]\( (-\infty, \infty) \)[/tex].
Out of the given multiple choice answers:
1. [tex]\( (-\infty, -4] \)[/tex]
2. [tex]\( (-\infty, \infty) \)[/tex]
3. [tex]\( [-4, 3] \)[/tex]
4. [tex]\( [3, \infty) \)[/tex]
The correct choice is:
(2) [tex]\( (-\infty, \infty) \)[/tex]
So, the domain of the function [tex]\( f(x) = x^2 + x - 12 \)[/tex] is all real numbers.
