To determine the range of the function [tex]\( y = \sqrt[3]{x + 8} \)[/tex], we must consider the properties of a cube root (or cubic root) function.
A cube root function is defined as [tex]\( y = \sqrt[3]{u} \)[/tex], where [tex]\( u = x + 8 \)[/tex] in our given function. Let’s analyze the behavior of this function step-by-step:
1. Cube Root Function Characteristics:
- The cube root function [tex]\( y = \sqrt[3]{u} \)[/tex] is defined for all real numbers [tex]\( u \)[/tex].
- It is an odd function, meaning that it possesses symmetry about the origin.
- The cube root of a positive number is positive, the cube root of zero is zero, and the cube root of a negative number is negative.
2. Domain of u:
- The domain of [tex]\( x \)[/tex] in the expression [tex]\( x + 8 \)[/tex] is all real numbers because any real number can be added to 8.
3. Range Determination:
- Since [tex]\( u = x + 8 \)[/tex] can take any real value (since [tex]\( x \)[/tex] can be any real number), the variable [tex]\( u \)[/tex] essentially spans from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex].
- The cube root function [tex]\( y = \sqrt[3]{u} \)[/tex] spans all real values as well because for every real number [tex]\( u \)[/tex], there exists a corresponding real number [tex]\( y \)[/tex].
4. Conclusion:
- Therefore, the range of [tex]\( y = \sqrt[3]{x + 8} \)[/tex] is all real numbers.
Thus, the range of the function [tex]\( y = \sqrt[3]{x + 8} \)[/tex] is [tex]\(-\infty < y < \infty\)[/tex].
Hence, the correct answer is:
[tex]\[
-\infty < y < \infty
\][/tex]
