To determine the range of the function [tex]\( y = \sqrt[3]{x + 8} \)[/tex], we'll consider the characteristics of cube root functions.
1. Understanding the Cube Root Function:
- The cube root function, [tex]\( y = \sqrt[3]{x} \)[/tex], is defined for all real numbers [tex]\( x \)[/tex].
- The cube root of any real number can produce any real number as the output. In other words, the cube root function is capable of taking on any value over the entire set of real numbers.
2. Transforming the Function:
- The given function is [tex]\( y = \sqrt[3]{x + 8} \)[/tex], which is a horizontal shift of the basic cube root function [tex]\( y = \sqrt[3]{x} \)[/tex] to the left by 8 units.
- Despite this shift, the fundamental behavior of the cube root function remains unchanged. It is still defined for all real numbers [tex]\( x \)[/tex] and can still produce any real number [tex]\( y \)[/tex] as the output.
3. Determining the Range:
- Since the cube root function can take any real number as output, shifting it horizontally does not change this property.
- Therefore, the range of [tex]\( y = \sqrt[3]{x + 8} \)[/tex] continues to be all real numbers.
Thus, the correct range of the function [tex]\( y = \sqrt[3]{x + 8} \)[/tex] is:
[tex]\[ -\infty < y < \infty \][/tex]
Therefore, the correct answer is:
[tex]\[ -\infty < y < \infty \][/tex]
