To determine the domain of the function [tex]\( f(x) = \sqrt{14 - 2x} \)[/tex], we need to ensure that the expression inside the square root is non-negative, because the square root of a negative number is not defined in the set of real numbers.
Let's set up the inequality to find when the inside of the square root is greater than or equal to zero:
[tex]\[ 14 - 2x \geq 0. \][/tex]
First, solve this inequality step-by-step:
1. Subtract 14 from both sides:
[tex]\[ -2x \geq -14. \][/tex]
2. Divide both sides by -2:
Remember that when we divide or multiply an inequality by a negative number, the inequality sign must be reversed.
[tex]\[ x \leq 7. \][/tex]
So, the solution to this inequality is [tex]\( x \leq 7 \)[/tex].
Therefore, the domain of the function [tex]\( f(x) = \sqrt{14 - 2x} \)[/tex] is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is less than or equal to 7.
In interval notation, the domain is:
[tex]\[ (-\infty, 7]. \][/tex]
Thus, the domain of [tex]\( f \)[/tex] is [tex]\( \boxed{(-\infty, 7]} \)[/tex].
