To determine the domain of the function [tex]\( f(x) = x^2 - 6x + 1 \)[/tex], we need to consider the nature of the function and any restrictions that might limit the values [tex]\( x \)[/tex] can take.
1. Type of Function:
[tex]\( f(x) = x^2 - 6x + 1 \)[/tex] is a quadratic function. A general quadratic function is given by [tex]\( ax^2 + bx + c \)[/tex] where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
2. Quadratic Functions:
For quadratic functions, the domain is worth considering because it represents all the possible values that [tex]\( x \)[/tex] can take. Quadratic functions are polynomial functions, and polynomial functions do not have restrictions like divisions by zero or square roots of negative numbers, which are common in other types of functions.
3. Identifying any Restrictions:
Since quadratic functions are defined for all [tex]\( x \)[/tex] (i.e., any real number), there are no restrictions like discontinuities, asymptotes, or undefined points.
4. Conclusion:
As there are no conditions or operations in [tex]\( f(x) \)[/tex] that impose any restrictions on [tex]\( x \)[/tex], the domain of [tex]\( f(x) \)[/tex] is all real numbers.
Therefore, the domain of [tex]\( f(x) = x^2 - 6x + 1 \)[/tex] is all real numbers, which is represented mathematically as:
[tex]\[
\boxed{\text{All real numbers}}
\][/tex]
