To determine the domain of the function [tex]\( f(x) = \sqrt{4x + 9} + 2 \)[/tex], we need to ensure that all operations within the function produce defined values for real numbers [tex]\( x \)[/tex].
1. Square Root Consideration:
The main point to consider is the square root function [tex]\( \sqrt{4x + 9} \)[/tex]. The expression inside the square root, [tex]\( 4x + 9 \)[/tex], must be non-negative because the square root of a negative number is not defined in the set of real numbers. Therefore, we need:
[tex]\[
4x + 9 \geq 0
\][/tex]
2. Solving the Inequality:
To find the values of [tex]\( x \)[/tex] that satisfy this inequality, we can solve it step-by-step:
[tex]\[
4x + 9 \geq 0
\][/tex]
Subtract 9 from both sides:
[tex]\[
4x \geq -9
\][/tex]
Divide both sides by 4:
[tex]\[
x \geq -\frac{9}{4}
\][/tex]
Hence, the inequality that can be used to determine the domain of the function [tex]\( f(x) = \sqrt{4x + 9} + 2 \)[/tex] is:
[tex]\[
4x + 9 \geq 0
\][/tex]
