To determine the domain of the function [tex]\( f(x) = 5x^6 \)[/tex], we need to identify the set of all possible values of [tex]\( x \)[/tex] for which the function is defined.
1. Check for Polynomial Functions:
The given function [tex]\( f(x) = 5x^6 \)[/tex] is a polynomial function. Polynomial functions are defined for all real numbers because there are no restrictions like division by zero or taking the square root of a negative number which would limit their domain.
2. Analyzing Restrictions:
- There are no fractions where the denominator could be zero.
- There are no logarithms that require positive arguments.
- There are no even roots (like square roots) that require non-negative arguments.
Since none of these restrictions are present in the function [tex]\( f(x) = 5x^6 \)[/tex], we conclude that there are no restrictions on the values that [tex]\( x \)[/tex] can take.
Based on this analysis, the domain of the function [tex]\( f(x) = 5x^6 \)[/tex] is:
[tex]\[
\boxed{\text{all real numbers}}
\][/tex]
