Answer :
To find the range of the function [tex]\( y = \sqrt[3]{x+8} \)[/tex], let's analyze the function step-by-step.
1. Understanding the Function:
- The function is given as [tex]\( y = \sqrt[3]{x + 8} \)[/tex].
- This is a cube root function shifted horizontally by 8 units to the left.
2. Examining the Domain:
- The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers.
- Therefore, [tex]\( \sqrt[3]{x + 8} \)[/tex] is also defined for all real numbers [tex]\( x \)[/tex].
3. Determining the Range:
- The cube root function can produce any real number output.
- As [tex]\( x \)[/tex] ranges from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex], the output [tex]\( y = \sqrt[3]{x + 8} \)[/tex] also ranges from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex].
To conclude, the range of the function [tex]\( y = \sqrt[3]{x + 8} \)[/tex] is all real numbers. Expressed in interval notation, this is [tex]\((-\infty, \infty)\)[/tex].
Therefore, the correct answer is:
[tex]\[ -\infty < y < \infty \][/tex]
1. Understanding the Function:
- The function is given as [tex]\( y = \sqrt[3]{x + 8} \)[/tex].
- This is a cube root function shifted horizontally by 8 units to the left.
2. Examining the Domain:
- The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers.
- Therefore, [tex]\( \sqrt[3]{x + 8} \)[/tex] is also defined for all real numbers [tex]\( x \)[/tex].
3. Determining the Range:
- The cube root function can produce any real number output.
- As [tex]\( x \)[/tex] ranges from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex], the output [tex]\( y = \sqrt[3]{x + 8} \)[/tex] also ranges from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex].
To conclude, the range of the function [tex]\( y = \sqrt[3]{x + 8} \)[/tex] is all real numbers. Expressed in interval notation, this is [tex]\((-\infty, \infty)\)[/tex].
Therefore, the correct answer is:
[tex]\[ -\infty < y < \infty \][/tex]
