What Is The Domain Of The Function $y=\sqrt[3]{x-1}$?A. − ∞ \textless X \textless Π -\infty \ \textless \ X \ \textless \ \pi − ∞ \textless X \textless Π B. − 1 \textless X \textless X -1 \ \textless \ X \ \textless \ X − 1 \textless X \textless X C. 0 0 0 D. 1 ≥ X × 3 1 \geq X \times 3 1 ≥ X × 3

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Understanding the Concept of Domain

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 or undefined results. In this article, we will explore the domain of the function $y=\sqrt[3]{x-1}$.

The Cubic Root Function

The function $y=\sqrt[3]{x-1}$ is a cubic root function, which means that it involves taking the cube root of the expression x-1. The cube root of a number is a value that, when multiplied by itself twice (or cubed), gives the original number. For example, the cube root of 27 is 3, because 3 multiplied by itself twice (333) equals 27.

Properties of the Cubic Root Function

One of the key properties of the cubic root function is that it is defined for all real numbers. This means that no matter what value of x we choose, the expression x-1 will always be a real number, and therefore the cube root of x-1 will also be a real number. This is in contrast to the square root function, which is only defined for non-negative real numbers.

Finding the Domain of the Function

To find the domain of the function $y=\sqrt[3]{x-1}$, we need to consider the values of x for which the expression x-1 is defined. Since the cube root function is defined for all real numbers, the only restriction on the domain of the function is that x-1 must be a real number. This means that x can be any real number, except for the value that would make x-1 equal to zero.

Solving for the Domain

To find the value of x that would make x-1 equal to zero, we can set up the equation x-1=0 and solve for x. This gives us x=1. Therefore, the only value of x that is not in the domain of the function is x=1.

Writing the Domain in Interval Notation

The domain of the function $y=\sqrt[3]{x-1}$ can be written in interval notation as (-∞, 1) ∪ (1, ∞). This means that the function is defined for all real numbers except for x=1.

Conclusion

In conclusion, the domain of the function $y=\sqrt[3]{x-1}$ is all real numbers except for x=1. This can be written in interval notation as (-∞, 1) ∪ (1, ∞). The function is defined for all real numbers, except for the value that would make x-1 equal to zero.

Frequently Asked Questions

  • Q: What is the domain of the function $y=\sqrt[3]{x-1}$? A: The domain of the function is all real numbers except for x=1.
  • Q: Why is the function not defined for x=1? A: The function is not defined for x=1 because x-1 would equal zero, and the cube root of zero is undefined.
  • Q: Can the function be defined for x=1? A: No, the function cannot be defined for x=1 because it would involve taking the cube root of zero, which is undefined.

Final Answer

The final answer is: (−∞,1)∪(1,∞)\boxed{(-\infty, 1) \cup (1, \infty)}

Understanding the Domain of the Function

In our previous article, we explored the domain of the function $y=\sqrt[3]{x-1}$. The domain of a function is the set of all possible input values for which the function is defined. In this article, we will answer some frequently asked questions about the domain of the function $y=\sqrt[3]{x-1}$.

Q&A

Q: What is the domain of the function $y=\sqrt[3]{x-1}$?

A: The domain of the function is all real numbers except for x=1. This can be written in interval notation as (-∞, 1) ∪ (1, ∞).

Q: Why is the function not defined for x=1?

A: The function is not defined for x=1 because x-1 would equal zero, and the cube root of zero is undefined.

Q: Can the function be defined for x=1?

A: No, the function cannot be defined for x=1 because it would involve taking the cube root of zero, which is undefined.

Q: What happens if I try to plug in x=1 into the function?

A: If you try to plug in x=1 into the function, you will get an undefined result. This is because the cube root of zero is undefined.

Q: Is the function defined for all real numbers except for x=1?

A: Yes, the function is defined for all real numbers except for x=1. This means that the function is defined for all real numbers, except for the value that would make x-1 equal to zero.

Q: Can I write the domain of the function in a different notation?

A: Yes, you can write the domain of the function in a different notation. For example, you can write it as (-∞, 1) ∪ (1, ∞), or as (-∞, 1) ∪ (1, ∞), or as (-∞, 1) ∪ (1, ∞).

Q: Is the domain of the function the same as the range of the function?

A: No, the domain of the function is not the same as the range of the function. The domain of the function is the set of all possible input values for which the function is defined, while the range of the function is the set of all possible output values.

Q: Can I find the domain of the function by looking at the graph of the function?

A: Yes, you can find the domain of the function by looking at the graph of the function. The domain of the function is the set of all x-values for which the function is defined.

Conclusion

In conclusion, the domain of the function $y=\sqrt[3]{x-1}$ is all real numbers except for x=1. This can be written in interval notation as (-∞, 1) ∪ (1, ∞). The function is defined for all real numbers, except for the value that would make x-1 equal to zero.

Final Answer

The final answer is: (−∞,1)∪(1,∞)\boxed{(-\infty, 1) \cup (1, \infty)}

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