Alright, let's delve into this problem step by step with the constraints provided.
To begin with, the function [tex]\( y = x^3 \)[/tex] is a cubic function. By definition, the domain of a cubic function is all real numbers, which means that [tex]\( x \)[/tex] can take any real value from [tex]\(-\infty\)[/tex] to [tex]\(+\infty\)[/tex].
However, the problem provides additional constraints that alter the domain:
1. Constraint 1: [tex]\( x > 0 \)[/tex]
This constraint specifies that [tex]\( x \)[/tex] must be greater than 0. Consequently, the domain of [tex]\( y = x^3 \)[/tex] under this constraint is all positive real numbers. This can be expressed as:
[tex]\[
x > 0
\][/tex]
2. Constraint 2: [tex]\( x \geq 0 \)[/tex]
This constraint specifies that [tex]\( x \)[/tex] must be greater than or equal to 0. Under this constraint, the domain of [tex]\( y = x^3 \)[/tex] includes all non-negative real numbers. This can be expressed as:
[tex]\[
x \geq 0
\][/tex]
To summarize:
- The general domain of [tex]\( y = x^3 \)[/tex] is all real numbers [tex]\( (-\infty, +\infty) \)[/tex].
- With the constraint [tex]\( x > 0 \)[/tex], the domain is [tex]\( x > 0 \)[/tex].
- With the constraint [tex]\( x \geq 0 \)[/tex], the domain is [tex]\( x \geq 0 \)[/tex].
Therefore, under the given constraints, the specific domains are:
- For [tex]\( x > 0 \)[/tex]: [tex]\( x > 0 \)[/tex]
- For [tex]\( x \geq 0 \)[/tex]: [tex]\( x \geq 0 \)[/tex]
These constraints adjust the otherwise infinite domain of the function to more restricted ranges.
