For the function [tex]W(x)=\sqrt[3]{7 x+23}[/tex], determine the domain. Write the domain in interval notation. [Do not use any decimals]
Domain of [tex]W(x)[/tex] : $\square$
Answer (1)
The function is W ( x ) = 3 7 x + 23 .
Cube roots are defined for all real numbers.
Therefore, 7 x + 23 can be any real number.
The domain of W ( x ) is ( − ∞ , ∞ ) .
Explanation
Understanding the Problem We are asked to find the domain of the function W ( x ) = 3 7 x + 23 . 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.
Implication for the Expression Therefore, 7 x + 23 can be any real number. This means there are no restrictions on the values that x can take.
Expressing as an Inequality To confirm this, we can express this as an inequality: − ∞ < 7 x + 23 < ∞
Isolating x Solving this inequality for x , we first subtract 23 from all parts of the inequality: − ∞ − 23 < 7 x < ∞ − 23 − ∞ < 7 x < ∞
Final Inequality Next, we divide all parts of the inequality by 7: − 7 ∞ < x < 7 ∞ − ∞ < x < ∞
Domain in Interval Notation This result tells us that x can be any real number. In interval notation, this is written as ( − ∞ , ∞ ) .
Final Answer Therefore, the domain of the function W ( x ) = 3 7 x + 23 is all real numbers, which in interval notation is ( − ∞ , ∞ ) .
Examples
Understanding the domain of a cube root function is useful in various real-world scenarios. For example, in physics, if you are modeling the volume of a cube as a function of some variable x and the relationship involves a cube root, knowing the domain helps you determine the valid range of x values for which the volume is physically meaningful. Similarly, in engineering, when designing systems involving fluid dynamics or heat transfer, cube root functions might appear, and understanding their domains ensures that the models are valid for the given physical constraints.
