To determine the range of a vertical translation of the function [tex]\( y = \sqrt[3]{x} \)[/tex], we need to understand both the nature of the original function and how vertical translations affect it.
1. Original Function [tex]\( y = \sqrt[3]{x} \)[/tex]:
- The cube root function, [tex]\( y = \sqrt[3]{x} \)[/tex], is defined for all real numbers [tex]\( x \)[/tex].
- This means that for any real number [tex]\( y \)[/tex], there is some real number [tex]\( x \)[/tex] such that [tex]\( y = \sqrt[3]{x} \)[/tex].
- Therefore, the range of [tex]\( y = \sqrt[3]{x} \)[/tex] is all real numbers, i.e., [tex]\(\{ y \mid y \text{ is a real number}\}\)[/tex].
2. Vertical Translation:
- A vertical translation shifts the graph of a function up or down without changing its basic shape.
- Mathematically, a vertical translation is of the form [tex]\( y = \sqrt[3]{x} + k \)[/tex], where [tex]\( k \)[/tex] is a constant.
- Shifting the function [tex]\( y = \sqrt[3]{x} \)[/tex] vertically by [tex]\( k \)[/tex] units still allows [tex]\( y \)[/tex] to take any real value, because for every [tex]\( y \)[/tex] in the original function, [tex]\( y - k \)[/tex] would also cover all real values.
Hence, the range of the translated function remains unchanged and includes all real numbers.
Therefore, the range of a vertical translation of [tex]\( y = \sqrt[3]{x} \)[/tex] is:
[tex]\(\{ y \mid y \text{ is a real number} \}\)[/tex].
