To determine the domain of the function \( y = x^3 \), let's analyze the properties of this function.
### Definition and Properties
The function \( y = x^3 \) is a cubic function. A cubic function is defined by a polynomial of degree three, and it can be expressed as:
[tex]\[ y = x^3. \][/tex]
### Understanding the Domain
The domain of a function is the set of all possible input values (x-values) that the function can accept without causing any mathematical errors or undefined behavior.
1. Exponentiation with Integers:
- For any real number \(x\):
[tex]\[ x^3 = x \cdot x \cdot x \][/tex]
- Cubing a real number means multiplying that number by itself three times.
2. Characteristics of Cubing:
- There are no restrictions on the values that \(x\) can take for \( y = x^3 \).
- \( x \) can be any real number: positive, negative, or zero.
### Conclusion
Since any real number squared and then multiplied by the original number will produce another real number, there are no limitations on the input values for \( x \) in the function \( y = x^3 \).
Thus, the domain of \( y = x^3 \) is all real numbers.
In set notation, this is expressed as:
[tex]\[ \text{Domain} = \{ x \in \mathbb{R} \} \][/tex]
or simply stated,
[tex]\[ \text{Domain} = \text{all real numbers}. \][/tex]
So, the correct answer is:
[tex]\[ \text{all real numbers} \][/tex]
