To find the domain of [tex]\( f(x) = \sqrt{4x + 9} + 2 \)[/tex], we need to ensure that the expression under the square root is non-negative. The square root function, [tex]\(\sqrt{y}\)[/tex], is only defined for [tex]\( y \geq 0 \)[/tex].
Let's start by examining the expression under the square root in the function [tex]\( f(x) \)[/tex]:
[tex]\[ \sqrt{4x + 9} \][/tex]
For this expression to be defined, the expression inside the square root must be non-negative:
[tex]\[ 4x + 9 \geq 0 \][/tex]
This is the inequality that ensures the square root is defined. Solving this inequality for [tex]\( x \)[/tex]:
1. First, isolate [tex]\( x \)[/tex] by subtracting 9 from both sides:
[tex]\[ 4x \geq -9 \][/tex]
2. Then, divide both sides by 4:
[tex]\[ x \geq -\frac{9}{4} \][/tex]
Therefore, the inequality that can be used to find the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ 4x + 9 \geq 0 \][/tex]
Thus, among the given options, the correct inequality is:
[tex]\[ 4 x + 9 \geq 0 \][/tex]
This inequality ensures that the expression under the square root is non-negative, and hence, the function [tex]\( f(x) = \sqrt{4x + 9} + 2 \)[/tex] is well-defined.
