To determine the range of the function [tex]\( y = \sqrt[3]{x+8} \)[/tex], let's understand what this function represents and how it behaves.
1. Understanding the Cube Root Function:
- The cube root function [tex]\( y = \sqrt[3]{x} \)[/tex] can take any real value for [tex]\( x \)[/tex] because it is defined for all real numbers.
- The cube root of a number can be positive, negative, or zero depending on the value of [tex]\( x \)[/tex].
2. Applying the Transformation:
- In the given function [tex]\( y = \sqrt[3]{x + 8} \)[/tex], there is a horizontal shift. Specifically, [tex]\( x \)[/tex] is replaced by [tex]\( x + 8 \)[/tex].
- This transformation shifts the entire graph of the cube root function [tex]\( y = \sqrt[3]{x} \)[/tex] to the left by 8 units.
3. Analyzing the Range:
- Since the cube root function [tex]\( y = \sqrt[3]{x} \)[/tex] is defined for all real [tex]\( x \)[/tex] and can yield any real number as [tex]\( y \)[/tex], the transformation [tex]\( x + 8 \)[/tex] does not change the overall behavior of the function.
- Thus, [tex]\( y = \sqrt[3]{x + 8} \)[/tex] will still be able to take any real value. As [tex]\( x \)[/tex] ranges from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex], [tex]\( y \)[/tex] will also range from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex].
4. Final Conclusion:
- After substituting various values of [tex]\( x \)[/tex] (real numbers from negative to positive infinity) and evaluating [tex]\( y \)[/tex], we see that [tex]\( y \)[/tex] can be any real number, from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex].
Therefore, the range of the function [tex]\( y = \sqrt[3]{x + 8} \)[/tex] is [tex]\( -\infty < y < \infty \)[/tex].
The correct answer is:
[tex]\[ -\infty < y < \infty \][/tex]
