To determine the range of the function [tex]\( y = \sqrt[3]{x+8} \)[/tex], we need to consider how the output [tex]\( y \)[/tex] behaves for all possible input values [tex]\( x \)[/tex].
1. Understanding the cube root function: The cube root function [tex]\( y = \sqrt[3]{x} \)[/tex] is defined for all real numbers. This means [tex]\( \sqrt[3]{x} \)[/tex] can take any real number as input and produce any real number as output.
2. Transforming the input: In the function [tex]\( y = \sqrt[3]{x+8} \)[/tex], you are simply shifting the input of the cube root function by 8 units to the left, which does not restrict the inputs or outputs. Therefore, for any real number [tex]\( x \)[/tex], [tex]\( x + 8 \)[/tex] is still a real number, and thus [tex]\( \sqrt[3]{x+8} \)[/tex] is well-defined for all [tex]\( x \)[/tex].
3. Range analysis: Since [tex]\( \sqrt[3]{x+8} \)[/tex] applies the cube root to [tex]\( x+8 \)[/tex], and the cube root function can output all real numbers, [tex]\( y \)[/tex] can indeed take any real value. This means the range is not restricted in any way and [tex]\( y \)[/tex] can be any real number.
Therefore, the range of the function [tex]\( y = \sqrt[3]{x+8} \)[/tex] is [tex]\( -\infty < y < \infty \)[/tex], or in other words, all real numbers.
Thus, the correct answer is:
[tex]\[ -\infty < y < \infty \][/tex]
