To find the domain of the square root function [tex]\( f(x) = \sqrt{7x - 42} \)[/tex], we need to ensure that the expression inside the square root [tex]\( 7x - 42 \)[/tex] is non-negative. This is because the square root of a negative number is not defined in the set of real numbers.
The inequality to solve is:
[tex]\[ 7x - 42 \geq 0 \][/tex]
Let's solve this step by step:
1. Start by isolating the term with [tex]\( x \)[/tex] on one side of the inequality:
[tex]\[ 7x - 42 \geq 0 \][/tex]
2. Add 42 to both sides of the inequality to move the constant term to the right side:
[tex]\[ 7x \geq 42 \][/tex]
3. Finally, divide by 7 to solve for [tex]\( x \)[/tex]:
[tex]\[ x \geq \frac{42}{7} \][/tex]
[tex]\[ x \geq 6 \][/tex]
Therefore, the domain of the function [tex]\( f(x) = \sqrt{7x - 42} \)[/tex] is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \geq 6 \)[/tex].
In interval notation, the domain is:
[tex]\[ [6, \infty) \][/tex]
Thus, the answer is:
[tex]\[ x \geq 6 \][/tex]
