What Is The Domain Of The Function Y = X − 1 3 Y=\sqrt[3]{x-1} Y = 3 X − 1 ​ ?A. − ∞ \textless X \textless ∞ -\infty\ \textless \ X\ \textless \ \infty − ∞ \textless X \textless ∞ B. − 1 \textless X \textless ∞ -1\ \textless \ X\ \textless \ \infty − 1 \textless X \textless ∞ C. 0 ≤ X \textless ∞ 0 \leq X\ \textless \ \infty 0 ≤ X \textless ∞ D. $1 \leq X\ \textless \

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Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of the variable (in this case, $x$) that can be plugged into the function without causing any problems. In this article, we will explore the concept of the domain of a function and use the specific example of $y=\sqrt[3]{x-1}$ to illustrate the process.

What is the Domain of a Function?

The domain of a function is a critical concept in mathematics, as it determines the set of all possible input values for which the function is defined. In general, the domain of a function is the set of all real numbers for which the function is defined. However, there are certain functions that are not defined for all real numbers, and these functions have a restricted domain.

Cubic Root Function

The cubic root function, denoted by $\sqrt[3]{x}$, is a function that takes a real number $x$ as input and returns the cube root of $x$. The cube root of a number is a number that, when multiplied by itself twice, gives the original number. For example, the cube root of $8$ is $2$, because $2 \times 2 \times 2 = 8$.

Domain of the Cubic Root Function

The domain of the cubic root function is all real numbers, because the cube root of any real number is defined. In other words, for any real number $x$, the cube root of $x$ is also a real number.

Domain of the Function $y=\sqrt[3]{x-1}$

Now that we have a good understanding of the cubic root function and its domain, let's consider the specific function $y=\sqrt[3]{x-1}$. This function takes a real number $x$ as input and returns the cube root of $x-1$. To determine the domain of this function, we need to consider the values of $x$ for which the expression $x-1$ is defined.

Restrictions on the Domain

The expression $x-1$ is defined for all real numbers $x$, because subtracting $1$ from any real number results in another real number. However, we need to consider the values of $x$ for which the cube root of $x-1$ is defined. Since the cube root function is defined for all real numbers, the only restriction on the domain of the function $y=\sqrt[3]{x-1}$ is that $x-1$ must be greater than or equal to $0$.

Solving the Inequality

To find the values of $x$ for which $x-1 \geq 0$, we can solve the inequality $x-1 \geq 0$. Adding $1$ to both sides of the inequality, we get $x \geq 1$.

Conclusion

In conclusion, the domain of the function $y=\sqrt[3]{x-1}$ is all real numbers $x$ such that $x \geq 1$. This means that the function is defined for all values of $x$ greater than or equal to $1$, and it is not defined for any values of $x$ less than $1$.

Answer

The correct answer is:

• B. $-1\ \textless \ x\ \textless \ \infty$

This answer is correct because the function $y=\sqrt[3]{x-1}$ is defined for all real numbers $x$ such that $x \geq 1$, which is equivalent to $-1\ \textless \ x\ \textless \ \infty$.

Final Thoughts

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Introduction

In our previous article, we explored the concept of the domain of a function and used the specific example of $y=\sqrt[3]{x-1}$ to illustrate the process. In this article, we will provide a Q&A guide to help you better understand the domain of a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of the variable (in this case, $x$) that can be plugged into the function without causing any problems.

Q: Why is the domain of a function important?

A: The domain of a function is important because it determines the set of all possible input values for which the function is defined. If the domain of a function is not properly defined, it can lead to incorrect results or even undefined behavior.

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

A: To determine the domain of a function, you need to consider the values of the input variable (in this case, $x$) for which the function is defined. You can do this by analyzing the function and identifying any restrictions on the input variable.

Q: What are some common restrictions on the domain of a function?

A: Some common restrictions on the domain of a function include:

• Division by zero: If a function involves division by zero, it is not defined for that value of the input variable.
• Square root of a negative number: If a function involves the square root of a negative number, it is not defined for that value of the input variable.
• Logarithm of a non-positive number: If a function involves the logarithm of a non-positive number, it is not defined for that value of the input variable.

Q: How do I handle restrictions on the domain of a function?

A: To handle restrictions on the domain of a function, you need to identify the values of the input variable for which the function is not defined and exclude them from the domain. You can do this by using inequalities or other mathematical expressions to describe the domain.

Q: Can the domain of a function be empty?

A: Yes, the domain of a function can be empty. This means that the function is not defined for any values of the input variable.

Q: Can the domain of a function be all real numbers?

A: Yes, the domain of a function can be all real numbers. This means that the function is defined for all values of the input variable.

Q: How do I determine the domain of a function with multiple variables?

A: To determine the domain of a function with multiple variables, you need to consider the values of all the input variables for which the function is defined. You can do this by analyzing the function and identifying any restrictions on the input variables.

Q: Can the domain of a function change depending on the input values?

A: Yes, the domain of a function can change depending on the input values. This means that the function may be defined for some values of the input variable but not others.

Q: How do I graph a function with a restricted domain?

A: To graph a function with a restricted domain, you need to identify the values of the input variable for which the function is not defined and exclude them from the graph. You can do this by using inequalities or other mathematical expressions to describe the domain.

Conclusion

In conclusion, the domain of a function is a critical concept in mathematics that determines the set of all possible input values for which the function is defined. By understanding the domain of a function, you can better analyze and graph the function, and make more informed decisions about the function's behavior. We hope this Q&A guide has helped you better understand the domain of a function.

Frequently Asked Questions

• Q: What is the domain of the function $y=\sqrt{x}$? A: The domain of the function $y=\sqrt{x}$ is all real numbers $x$ such that $x \geq 0$.
• Q: What is the domain of the function $y=\frac{1}{x}$? A: The domain of the function $y=\frac{1}{x}$ is all real numbers $x$ such that $x \neq 0$.
• Q: What is the domain of the function $y=\log(x)$? A: The domain of the function $y=\log(x)$ is all real numbers $x$ such that $x > 0$.
• Q: What is the domain of the function $y=\sin(x)$? A: The domain of the function $y=\sin(x)$ is all real numbers $x$.

Additional Resources

• Mathematics textbooks: For a comprehensive understanding of the domain of a function, we recommend consulting a mathematics textbook.
• Online resources: For additional resources and practice problems, we recommend visiting online resources such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
• Mathematics communities: For discussion and support, we recommend joining online mathematics communities such as Reddit's r/learnmath and Stack Exchange's Mathematics community.