Understanding the Range of the Function

Oct 13, 2025 by talynt.org 91 views

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 radicals, such as the cubic root function, understanding the range is crucial to determine the possible output values. In this article, we will explore the range of the function $y=\sqrt[3]{x+8}$ and discuss the possible answer choices.

What is the Cubic Root Function?

The cubic root function, denoted as $\sqrt[3]{x}$ , is a mathematical function that returns the cube root of a given number $x$ . In other words, it is the inverse operation of cubing a number. The cubic root function is defined for all real numbers, and its graph is a curve that passes through the origin $(0,0)$ .

The Function $y=\sqrt[3]{x+8}$

The given function $y=\sqrt[3]{x+8}$ is a transformation of the cubic root function. The term $x+8$ inside the radical indicates that the function is shifted 8 units to the left. This means that the graph of the function will be the same as the graph of the cubic root function, but shifted 8 units to the left.

Understanding the Range of the Function

To determine the range of the function $y=\sqrt[3]{x+8}$ , we need to consider the possible values of $x$ and how they affect the output value $y$ . Since the function is a cubic root function, it is defined for all real numbers. However, the term $x+8$ inside the radical indicates that the function is shifted 8 units to the left. This means that the minimum value of $x$ is $-8$ , and the function will produce output values starting from $y=0$ .

Analyzing the Answer Choices

Now that we have a better understanding of the function $y=\sqrt[3]{x+8}$ , let's analyze the possible answer choices:

A. $-\infty\ \textless \ y\ \textless \ \infty$ : This answer choice suggests that the range of the function is all real numbers. However, we know that the function is shifted 8 units to the left, which means that the minimum value of $y$ is 0, not $-\infty$ .
B. $-8\ \textless \ y\ \textless \ \infty$ : This answer choice suggests that the range of the function is all real numbers greater than $-8$ . This is a possible answer, but we need to consider the minimum value of $y$ .
C. $0 \leq y\ \textless \ \infty$ : This answer choice suggests that the range of the function is all real numbers greater than or equal to 0. This is a possible answer, but we need to consider the minimum value of $y$ .
D. $2 \leq y\ \textless \ \infty$ : This answer choice suggests that the range of the function is all real numbers greater than or equal to 2. This is not a possible answer, as the minimum value of $y$ is 0, not 2.

Conclusion

In conclusion, the range of the function $y=\sqrt[3]{x+8}$ is all real numbers greater than or equal to 0. This means that the correct answer choice is C. $0 \leq y\ \textless \ \infty$ .

Final Answer

Q: What is the range of the function $y=\sqrt[3]{x+8}$ ?

A: The range of the function $y=\sqrt[3]{x+8}$ is all real numbers greater than or equal to 0.

Q: Why is the range of the function $y=\sqrt[3]{x+8}$ all real numbers greater than or equal to 0?

A: The range of the function $y=\sqrt[3]{x+8}$ is all real numbers greater than or equal to 0 because the function is a cubic root function, and the term $x+8$ inside the radical indicates that the function is shifted 8 units to the left. This means that the minimum value of $x$ is $-8$ , and the function will produce output values starting from $y=0$ .

Q: What is the significance of the term $x+8$ inside the radical?

A: The term $x+8$ inside the radical indicates that the function is shifted 8 units to the left. This means that the graph of the function will be the same as the graph of the cubic root function, but shifted 8 units to the left.

Q: How does the term $x+8$ affect the range of the function?

A: The term $x+8$ inside the radical affects the range of the function by shifting the minimum value of $x$ to $-8$ . This means that the function will produce output values starting from $y=0$ .

Q: What is the relationship between the range of the function $y=\sqrt[3]{x+8}$ and the answer choices?

A: The range of the function $y=\sqrt[3]{x+8}$ is related to the answer choices as follows:

A. $-\infty\ \textless \ y\ \textless \ \infty$ : This answer choice is incorrect because the range of the function is not all real numbers.
B. $-8\ \textless \ y\ \textless \ \infty$ : This answer choice is incorrect because the range of the function is not all real numbers greater than $-8$ .
C. $0 \leq y\ \textless \ \infty$ : This answer choice is correct because the range of the function is all real numbers greater than or equal to 0.
D. $2 \leq y\ \textless \ \infty$ : This answer choice is incorrect because the range of the function is not all real numbers greater than or equal to 2.

Q: What is the final answer to the problem?

A: The final answer to the problem is C. $0 \leq y\ \textless \ \infty$ .

Q: What is the significance of understanding the range of the function $y=\sqrt[3]{x+8}$ ?

A: Understanding the range of the function $y=\sqrt[3]{x+8}$ is significant because it helps to determine the possible output values of the function for a given input value. This is important in various mathematical and real-world applications, such as optimization problems, data analysis, and scientific modeling.

Q: How can I apply the concept of the range of a function to real-world problems?

A: You can apply the concept of the range of a function to real-world problems by considering the possible output values of a function for a given input value. This can help you to make informed decisions, optimize processes, and analyze data. For example, in economics, understanding the range of a function can help you to determine the possible prices of a product based on the cost of production and demand.