For the function [tex]Q(x)=\sqrt[3]{3 x+28}[/tex], determine the domain. Write the domain in interval notation. [Do not use any decimals]
Domain of [tex]Q(x)[/tex]: [ ]
Answer (1)
The function is Q ( x ) = 3 3 x + 28 .
Cube roots are defined for all real numbers, so 3 x + 28 can be any real number.
This means x can be any real number, or − ∞ < x < ∞ .
The domain in interval notation is ( − ∞ , ∞ ) .
Explanation
Understanding the Problem We are asked to find the domain of the function Q ( x ) = 3 3 x + 28 . The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Cube Root Property Since we are taking the cube root of an expression, the expression inside the cube root can be any real number. This is because we can take the cube root of any real number, whether it is positive, negative, or zero.
Analyzing the Expression Inside the Cube Root Therefore, 3 x + 28 can be any real number. This means there are no restrictions on the values that x can take. We can express this as − ∞ < 3 x + 28 < ∞ .
Solving for x To find the domain of x , we can solve the inequality. However, since 3 x + 28 can be any real number, we can directly say that x can be any real number. This simplifies to − ∞ < x < ∞ .
Expressing the Domain in Interval Notation In interval notation, the domain of Q ( x ) is ( − ∞ , ∞ ) .
Examples
Understanding the domain of functions like Q ( x ) = 3 3 x + 28 is crucial in many real-world applications. For instance, if x represents time and Q ( x ) represents the volume of a container, knowing the domain tells us for what time intervals the volume function is valid. Since the cube root function is defined for all real numbers, the volume can be calculated for any time x , making it a versatile model.
