To determine the domain of the function [tex]\( f(x) = \sqrt{4x + 9} + 2 \)[/tex], we need to ensure that the expression within the square root is non-negative (i.e., it must be greater than or equal to zero) since the square root of a negative number is not defined in the set of real numbers.
Let's analyze the expression under the square root:
[tex]\[ 4x + 9 \][/tex]
This expression must satisfy the following condition:
[tex]\[ 4x + 9 \geq 0 \][/tex]
Now, let's solve this inequality step-by-step to find the domain:
1. Isolate the term involving [tex]\( x \)[/tex]:
[tex]\[ 4x + 9 \geq 0 \][/tex]
2. Subtract 9 from both sides to isolate the [tex]\( 4x \)[/tex] term:
[tex]\[ 4x \geq -9 \][/tex]
3. Divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[ x \geq -\frac{9}{4} \][/tex]
So, the inequality that we need to satisfy to find the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ 4x + 9 \geq 0 \][/tex]
Thus, the correct inequality from the given options is:
[tex]\[ 4x + 9 \geq 0 \][/tex]
The domain of the function [tex]\( f(x) = \sqrt{4x + 9} + 2 \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \geq -\frac{9}{4} \)[/tex].
