To find the domain of the function [tex]\( v(x) = \sqrt{-7x + 42} \)[/tex], we need to ensure that the expression inside the square root is non-negative, as the square root of a negative number is not defined in the set of real numbers.
So, we start by setting up the inequality:
[tex]\[
-7x + 42 \geq 0
\][/tex]
To solve this inequality, follow these steps:
1. Isolate the variable term:
[tex]\[
-7x + 42 \geq 0
\][/tex]
Subtract 42 from both sides:
[tex]\[
-7x \geq -42
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
Divide both sides of the inequality by -7. Remember, dividing or multiplying an inequality by a negative number reverses the inequality sign:
[tex]\[
x \leq 6
\][/tex]
Thus, [tex]\( x \)[/tex] must be less than or equal to 6 for [tex]\( -7x + 42 \)[/tex] to be non-negative.
3. Write the domain in interval notation:
The domain includes all [tex]\( x \)[/tex] values less than or equal to 6, which can be expressed in interval notation as:
[tex]\[
(-\infty, 6]
\][/tex]
Therefore, the domain of the function [tex]\( v(x) = \sqrt{-7x + 42} \)[/tex] is [tex]\( (-\infty, 6] \)[/tex].
