Question
Which of the following is the range of the function \( g(x)=-\sqrt[3]{x+2}-5 \) ?
Option \#1: \( -\infty
Ask by Young Fowler. in the United States
Jan 23,2025
Ask by Young Fowler. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The best answer is Option #4: \( -\infty < y < \infty \).
Solution
The best answer is **Option #4: \( -\infty < y < \infty \)**.
This is because the function \( g(x) = -\sqrt[3]{x+2} - 5 \) can take any real value for \( y \) as \( x \) varies over all real numbers.
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Simplify this solution Bonus Knowledge
The function \( g(x) = -\sqrt[3]{x+2} - 5 \) transforms the basic cube root function, which itself has a range of all real numbers. Since it is flipped (due to the negative sign) and shifted downward by 5 units, the result is that the overall function can still take on any real number. Thus, the range remains \( -\infty < y < \infty \). To deepen your understanding, consider that cube root functions are unique because they can handle all inputs and output values without boundary restrictions. This means that when we manipulate them with transformations like shifts and reflections, we still observe the same freedom in the range—the landscape of possible outputs is as vast as it gets!
