To determine the domain of the square root function [tex]\( f(x) = \sqrt{6x - 24} + 5 \)[/tex], we need to ensure the expression inside the square root is non-negative. This is because the square root of a negative number is not defined within the set of real numbers.
Let's consider the inequality:
[tex]\[ 6x - 24 \geq 0 \][/tex]
First, isolate [tex]\( x \)[/tex] on one side of the inequality.
1. Add 24 to both sides:
[tex]\[ 6x - 24 + 24 \geq 0 + 24 \][/tex]
[tex]\[ 6x \geq 24 \][/tex]
2. Divide both sides by 6:
[tex]\[ \frac{6x}{6} \geq \frac{24}{6} \][/tex]
[tex]\[ x \geq 4 \][/tex]
Hence, [tex]\( x \)[/tex] must be at least 4 for the function [tex]\( f(x) \)[/tex] to be defined. Therefore, the domain of the function [tex]\( f(x) = \sqrt{6x - 24} + 5 \)[/tex] is:
[tex]\[ x \geq 4 \][/tex]
So, the domain is:
[tex]\[ x \geq 4 \][/tex]
