For the function [tex]$Q(x)=\sqrt{4 x+12}$[/tex], determine the domain. Write the domain in interval notation.
Answer (1)
The domain of Q ( x ) = 4 x + 12 requires 4 x + 12 ≥ 0 .
Solve the inequality 4 x + 12 ≥ 0 for x .
This gives x ≥ − 3 .
Express the solution in interval notation: [ − 3 , ∞ ) .
Explanation
Understanding the Problem We are given the function Q ( x ) = 4 x + 12 and we want to find its domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. Since we have a square root, the expression inside the square root must be greater than or equal to zero.
Setting up the Inequality To find the domain, we need to solve the inequality 4 x + 12 ≥ 0 .
Isolating the Variable Term Subtract 12 from both sides of the inequality: 4 x + 12 − 12 ≥ 0 − 12 4 x ≥ − 12
Solving for x Divide both sides by 4: 4 4 x ≥ 4 − 12 x ≥ − 3
Expressing the Solution in Interval Notation The solution to the inequality is x ≥ − 3 . This means that the domain of the function Q ( x ) is all real numbers greater than or equal to -3. In interval notation, this is written as [ − 3 , ∞ ) .
Final Answer Therefore, the domain of the function Q ( x ) = 4 x + 12 is [ − 3 , ∞ ) .
Examples
Consider a scenario where you are designing a garden and need to determine the minimum length of a side of a square plot such that the area is at least a certain value. If the area is given by A = ( x + 3 ) 2 , where x represents an additional length to a base length of -3 (to ensure positivity), finding the domain ensures that x is always a valid, non-negative value. This ensures the garden's area is a real, non-negative number, making the design practical and physically possible. Understanding domains helps in real-world applications where quantities must be real and non-negative.
