To determine the range of the function [tex]\( y = \sqrt[3]{x + 8} \)[/tex], let's analyze how this function behaves.
1. Understanding the cube root function:
- The cube root function [tex]\( \sqrt[3]{z} \)[/tex] is defined for all real numbers [tex]\( z \)[/tex]. This means that the cube root of any real number [tex]\( z \)[/tex] will result in a real number.
2. Transforming the argument:
- In our case, the function is [tex]\( y = \sqrt[3]{x + 8} \)[/tex]. Here, [tex]\( z = x + 8 \)[/tex].
3. Domain of [tex]\( x \)[/tex]:
- The expression [tex]\( x + 8 \)[/tex] can take any real value since [tex]\( x \)[/tex] is a real number. As [tex]\( x \)[/tex] spans all real numbers, so does [tex]\( x + 8 \)[/tex].
4. Behavior of [tex]\( y \)[/tex]:
- Since [tex]\( x + 8 \)[/tex] can be any real number, the cube root of [tex]\( x + 8 \)[/tex] will also output any real number. For example:
- If [tex]\( x \)[/tex] is very large positively, [tex]\( x + 8 \)[/tex] is also large, and [tex]\( \sqrt[3]{x + 8} \)[/tex] will be a large positive number.
- If [tex]\( x \)[/tex] is very large negatively, [tex]\( x + 8 \)[/tex] can still be very negative, and [tex]\( \sqrt[3]{x + 8} \)[/tex] will be a large negative number.
- If [tex]\( x+8 = 0 \)[/tex], then [tex]\( y = \sqrt[3]{0} = 0 \)[/tex].
Hence, the cube root function [tex]\( y = \sqrt[3]{x + 8} \)[/tex] can produce any real number as an output. Its range is all real numbers.
The correct answer is:
[tex]\[ -\infty < y < \infty \][/tex]
