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What is the range of the function [tex]y = \sqrt[3]{x+8}[/tex]?

A. [tex]-\infty \ \textless \ y \ \textless \ \infty[/tex]
B. [tex]-8 \ \textless \ y \ \textless \ \infty[/tex]
C. [tex]0 \leq y \ \textless \ \infty[/tex]
D. [tex]2 \leq y \ \textless \ \infty[/tex]

Sagot :

GinnyAnswer
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]
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