Answer :
To determine the domain of the function \( f(x) = \sqrt{4x + 9} + 2 \), we need to ensure that 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.
To do so, we solve the inequality:
[tex]\[ 4x + 9 \geq 0 \][/tex]
Let’s break down the solution step by step.
1. Write the inequality:
We need the term under the square root to be greater than or equal to zero.
[tex]\[ 4x + 9 \geq 0 \][/tex]
2. Isolate \( x \):
Subtract 9 from both sides of the inequality.
[tex]\[ 4x \geq -9 \][/tex]
3. Solve for \( x \):
Divide both sides by 4 to isolate \( x \).
[tex]\[ x \geq -\frac{9}{4} \][/tex]
Therefore, the inequality \( 4x + 9 \geq 0 \) allows us to determine the domain of the function. Hence, the correct inequality to use is:
[tex]\[ 4 x + 9 \geq 0 \][/tex]
To do so, we solve the inequality:
[tex]\[ 4x + 9 \geq 0 \][/tex]
Let’s break down the solution step by step.
1. Write the inequality:
We need the term under the square root to be greater than or equal to zero.
[tex]\[ 4x + 9 \geq 0 \][/tex]
2. Isolate \( x \):
Subtract 9 from both sides of the inequality.
[tex]\[ 4x \geq -9 \][/tex]
3. Solve for \( x \):
Divide both sides by 4 to isolate \( x \).
[tex]\[ x \geq -\frac{9}{4} \][/tex]
Therefore, the inequality \( 4x + 9 \geq 0 \) allows us to determine the domain of the function. Hence, the correct inequality to use is:
[tex]\[ 4 x + 9 \geq 0 \][/tex]
