Certainly! Let's find the value of [tex]\( y \)[/tex] with the given equation [tex]\( y = \sqrt{1 - 2x^3} \)[/tex]. Here is a step-by-step solution:
1. Understand the Expression Inside the Square Root:
The given equation is [tex]\( y = \sqrt{1 - 2x^3} \)[/tex]. The expression inside the square root is [tex]\( 1 - 2x^3 \)[/tex]. This expression will determine the value inside the square root.
2. Form the Function:
Functionally, we express [tex]\( y \)[/tex] as:
[tex]\[
y = \sqrt{1 - 2x^3}
\][/tex]
3. Ensure Valid Inputs:
Since we are dealing with a square root, the expression inside the square root, [tex]\( 1 - 2x^3 \)[/tex], must be non-negative because the square root of a negative number is not defined in the set of real numbers. Therefore, we need:
[tex]\[
1 - 2x^3 \geq 0
\][/tex]
4. Solve for the Domain:
Rearrange the inequality to find the domain of [tex]\( x \)[/tex]:
[tex]\[
1 \geq 2x^3 \implies \frac{1}{2} \geq x^3 \implies x^3 \leq \frac{1}{2}
\][/tex]
Therefore, the cube roots of both sides give the interval:
[tex]\[
x \leq \sqrt[3]{\frac{1}{2}}
\][/tex]
5. Conclusion on Function Validity:
The function [tex]\( y = \sqrt{1 - 2x^3} \)[/tex] is valid for any [tex]\( x \)[/tex] such that:
[tex]\[
x \leq \sqrt[3]{\frac{1}{2}}
\][/tex]
By following the steps above, we recognize that [tex]\( y = \sqrt{1 - 2x^3} \)[/tex], and it is defined for [tex]\( x \)[/tex] values that satisfy [tex]\( x \leq \sqrt[3]{\frac{1}{2}} \)[/tex].
Thus, [tex]\( y = \sqrt{1 - 2x^3} \)[/tex] provides the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] under the real number constraints specified so that the value inside the square root remains non-negative.
