To determine the range of the function [tex]\( y = \sqrt[3]{x + 8} \)[/tex], we need to understand how the function behaves as we vary the value of [tex]\( x \)[/tex].
### Step-by-Step Solution
1. Understand the Cubic Root Function:
- The function [tex]\( y = \sqrt[3]{x} \)[/tex] is defined for all real [tex]\( x \)[/tex]. This function can take any real value because the cubic root of a real number can be any real number. This means that we can plug in any real value for [tex]\( x \)[/tex] and get a real value for [tex]\( y \)[/tex].
2. Analyze the Transformation [tex]\( x + 8 \)[/tex]:
- The function given is [tex]\( y = \sqrt[3]{x + 8} \)[/tex]. This is a horizontal shift of the basic cubic root function [tex]\( y = \sqrt[3]{x} \)[/tex] to the left by 8 units, thanks to the [tex]\( x + 8 \)[/tex] inside the cubic root.
3. Determine the Range:
- Since the basic function [tex]\( y = \sqrt[3]{x} \)[/tex] has a range of all real numbers (i.e., [tex]\( -\infty < y < \infty \)[/tex]), and the horizontal shift does not affect the range, the function [tex]\( y = \sqrt[3]{x + 8} \)[/tex] still has a range of all real numbers.
Therefore, the range of the function [tex]\( y = \sqrt[3]{x + 8} \)[/tex] is:
[tex]\[ -\infty < y < \infty \][/tex]
So, the correct choice from the given options is:
[tex]\[ -\infty < y < \infty \][/tex]
