What Is The Range Of The Function $y=\sqrt[3]{x+8}$?A. − ∞ \textless Y \textless ∞ -\infty \ \textless \ Y \ \textless \ \infty − ∞ \textless Y \textless ∞ B. − 8 \textless Y \textless ∞ -8 \ \textless \ Y \ \textless \ \infty − 8 \textless Y \textless ∞ C. 0 ≤ Y \textless ∞ 0 \leq Y \ \textless \ \infty 0 ≤ Y \textless ∞ D. $2 \leq Y \

Mar 12, 2025 by ADMIN 351 views

**Understanding the Range of a Cubic Root Function**

Introduction

In mathematics, the range of a function is the set of all possible output values it can produce for the given input values. When dealing with functions involving cubic roots, it's essential to understand the behavior of the function and how it affects the range. In this article, we'll explore the range of the function $y=\sqrt[3]{x+8}$ and determine the correct answer from the given options.

What is a Cubic Root Function?

A cubic root function is a type of function that involves taking the cube root of an expression. The cube root of a number is a value that, when multiplied by itself twice (or cubed), gives the original number. The general form of a cubic root function is $y=\sqrt[3]{x+a}$, where $a$ is a constant.

Analyzing the Given Function

The given function is $y=\sqrt[3]{x+8}$. To understand the range of this function, we need to consider the behavior of the cube root function. The cube root function is an increasing function, meaning that as the input value increases, the output value also increases.

Finding the Minimum Value of the Function

To find the minimum value of the function, we need to find the value of $x$ that makes the expression inside the cube root equal to zero. This is because the cube root of zero is zero. Setting $x+8=0$ , we get $x=-8$ . Therefore, the minimum value of the function is $y=\sqrt[3]{-8+8}=\sqrt[3]{0}=0$ .

Determining the Range of the Function

Since the cube root function is increasing, the minimum value of the function is $y=0$ . As the input value $x$ increases, the output value $y$ also increases without bound. Therefore, the range of the function is all real numbers greater than or equal to zero, including zero.

Conclusion

In conclusion, the range of the function $y=\sqrt[3]{x+8}$ is $0 \leq y \ \textless \ \infty$ . This means that the function can produce any value greater than or equal to zero, but it cannot produce any negative values.

Answer

The correct answer is:

C. $0 \leq y \ \textless \ \infty$

Final Thoughts

Understanding the range of a function is crucial in mathematics, as it helps us determine the possible output values of a function for a given input. In this article, we explored the range of the function $y=\sqrt[3]{x+8}$ and determined that the correct answer is $0 \leq y \ \textless \ \infty$ . We hope this article has provided valuable insights into the behavior of cubic root functions and their ranges.

References

[1] "Cubic Root Function" by Math Open Reference. Retrieved from https://www.mathopenref.com/cubicroot.html
[2] "Range of a Function" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f6b7d9/x2f6b7d9-2-t/x2f6b7d9-2-t-a

Q: What is the range of a cubic root function?

A: The range of a cubic root function is all real numbers greater than or equal to zero, including zero. This means that the function can produce any value greater than or equal to zero, but it cannot produce any negative values.

Q: How do I find the minimum value of a cubic root function?

A: To find the minimum value of a cubic root function, you need to find the value of $x$ that makes the expression inside the cube root equal to zero. This is because the cube root of zero is zero. Setting $x+a=0$ , where $a$ is a constant, you can find the minimum value of the function.

Q: What happens to the output value of a cubic root function as the input value increases?

A: As the input value $x$ increases, the output value $y$ also increases without bound. This is because the cube root function is an increasing function.

Q: Can a cubic root function produce any negative values?

A: No, a cubic root function cannot produce any negative values. The cube root of a negative number is a complex number, but the range of a cubic root function is all real numbers greater than or equal to zero, including zero.

Q: How do I determine the range of a cubic root function?

A: To determine the range of a cubic root function, you need to consider the behavior of the cube root function. You can find the minimum value of the function by setting the expression inside the cube root equal to zero, and then determine the range of the function based on the behavior of the cube root function.

Q: What is the significance of the range of a cubic root function?

A: The range of a cubic root function is significant because it helps us determine the possible output values of the function for a given input. Understanding the range of a function is crucial in mathematics, as it helps us analyze the behavior of the function and make predictions about its output.

Q: Can I use the range of a cubic root function to solve equations involving the function?

A: Yes, you can use the range of a cubic root function to solve equations involving the function. By understanding the range of the function, you can determine the possible values of the input that satisfy the equation.

Q: How do I apply the range of a cubic root function to real-world problems?

A: You can apply the range of a cubic root function to real-world problems by using it to model and analyze real-world phenomena. For example, you can use the range of a cubic root function to model population growth, chemical reactions, and other phenomena that involve cubic root functions.

Q: What are some common applications of cubic root functions?

A: Some common applications of cubic root functions include:

Modeling population growth
Analyzing chemical reactions
Studying the behavior of complex systems
Solving equations involving cubic roots
Modeling real-world phenomena that involve cubic root functions

Q: Can I use the range of a cubic root function to determine the domain of the function?

A: Yes, you can use the range of a cubic root function to determine the domain of the function. By understanding the range of the function, you can determine the possible values of the input that satisfy the equation.

Q: How do I determine the domain of a cubic root function?

A: To determine the domain of a cubic root function, you need to consider the behavior of the cube root function. You can find the domain of the function by considering the values of $x$ that make the expression inside the cube root equal to zero or undefined.

Q: What is the relationship between the range and domain of a cubic root function?

A: The range and domain of a cubic root function are related in that the range of the function determines the possible output values, while the domain of the function determines the possible input values. Understanding the relationship between the range and domain of a function is crucial in mathematics, as it helps us analyze the behavior of the function and make predictions about its output.