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 (2)
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.
Solving for x Therefore, 7 x + 23 can be any real number. This means there are no restrictions on the values of x . We can write this as:
− ∞ < 7 x + 23 < ∞
To solve for x , we can subtract 23 from all parts of the inequality:
− ∞ − 23 < 7 x < ∞ − 23
− ∞ < 7 x < ∞
Now, we divide all parts of the inequality by 7:
− ∞/7 < x < ∞/7
− ∞ < x < ∞
This means that x can be any real number.
Expressing the Domain in Interval Notation In interval notation, the domain of W ( x ) is all real numbers, which is written as ( − ∞ , ∞ ) .
Final Answer Therefore, the domain of the function W ( x ) = 3 7 x + 23 is ( − ∞ , ∞ ) .
Examples
Understanding the domain of a function is crucial in many real-world applications. For instance, if x represents the number of hours a machine operates, and W ( x ) represents the amount of product produced, knowing the domain helps determine the feasible range of operating hours. If the domain is ( − ∞ , ∞ ) , it suggests the machine can theoretically operate for any number of hours. However, in reality, there might be physical limitations, such as the machine's lifespan or energy constraints, which would further restrict the practical domain.
The domain of the function W ( x ) = 3 7 x + 23 is all real numbers, which in interval notation is expressed as ( − ∞ , ∞ ) . This is because the cube root function can accept any real number as input. Therefore, there are no restrictions on the values of x .
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