Answer :
To determine the range of the function \( y = x^3 \), we need to analyze the behavior of the function over all possible values of \( x \).
### Step-by-Step Solution:
1. Understanding the Function:
The given function is \( y = x^3 \), which is a cubic function. Cubic functions have a distinct shape and tend to cover all values more extensively than quadratic or linear functions.
2. Increasing Nature:
The function \( y = x^3 \) is monotonic, which means it is consistently increasing or decreasing over its entire domain. Specifically, the cubic function \( y = x^3 \) increases without bound as \( x \) increases and decreases without bound as \( x \) decreases.
3. Domain:
The domain of the function \( y = x^3 \) includes all real numbers, \( (-\infty, \infty) \), since any real number substituted for \( x \) results in a real number \( y \).
4. Behavior at Extremes:
- As \( x \to \infty \): \( y = x^3 \to \infty \)
- As \( x \to -\infty \): \( y = x^3 \to -\infty \)
This indicates that as \( x \) takes on increasingly large positive or negative values, \( y \) also takes on increasingly large positive or negative values, respectively.
5. Covering All Real Values:
- For positive values of \( x \), \( y \) can be any positive real number.
- For negative values of \( x \), \( y \) can be any negative real number.
- For \( x = 0 \), \( y = 0^3 = 0 \).
Therefore, the output \( y \) of the function \( y = x^3 \) can take any real number value.
### Conclusion:
Given the above analysis, the range of the function \( y = x^3 \) is indeed all real numbers. In other words, \( y \) can be any real number, spanning from \( -\infty \) to \( \infty \).
Therefore, the correct answer is:
1. all real numbers
### Step-by-Step Solution:
1. Understanding the Function:
The given function is \( y = x^3 \), which is a cubic function. Cubic functions have a distinct shape and tend to cover all values more extensively than quadratic or linear functions.
2. Increasing Nature:
The function \( y = x^3 \) is monotonic, which means it is consistently increasing or decreasing over its entire domain. Specifically, the cubic function \( y = x^3 \) increases without bound as \( x \) increases and decreases without bound as \( x \) decreases.
3. Domain:
The domain of the function \( y = x^3 \) includes all real numbers, \( (-\infty, \infty) \), since any real number substituted for \( x \) results in a real number \( y \).
4. Behavior at Extremes:
- As \( x \to \infty \): \( y = x^3 \to \infty \)
- As \( x \to -\infty \): \( y = x^3 \to -\infty \)
This indicates that as \( x \) takes on increasingly large positive or negative values, \( y \) also takes on increasingly large positive or negative values, respectively.
5. Covering All Real Values:
- For positive values of \( x \), \( y \) can be any positive real number.
- For negative values of \( x \), \( y \) can be any negative real number.
- For \( x = 0 \), \( y = 0^3 = 0 \).
Therefore, the output \( y \) of the function \( y = x^3 \) can take any real number value.
### Conclusion:
Given the above analysis, the range of the function \( y = x^3 \) is indeed all real numbers. In other words, \( y \) can be any real number, spanning from \( -\infty \) to \( \infty \).
Therefore, the correct answer is:
1. all real numbers
