The graph of the cube root parent function was translated to create the graph f(x)=(x+6)−2f(x)=(x+6)−2, which is shown in the graph below.Which of the following represents the range of using interval f(x)f(x) notation?

A.(−4, 1)(−4, 1)

B.(−6, ∞)(−6, ∞)

C.[−4, 1][−4, 1]

D.(−∞, ∞)(−∞, ∞)


The given function is \( f(x) = \sqrt[3]{x+6} - 2 \), which is a translation of the cube root parent function. Let's analyze the transformations:

1. **Translation of \(x+6\)**: The \(+6\) inside the cube root shifts the graph **6 units to the left**.
2. **Translation of \(-2\)**: The \(-2\) outside the cube root shifts the graph **2 units down**.

### Parent Function:
The parent cube root function \( f(x) = \sqrt[3]{x} \) has a range of \( (-\infty, \infty) \) because cube roots can take any real value as \(x\) varies.

### Transformed Function:
- The range of the function remains **unchanged** because no transformation affects the range in such a way to restrict it. Shifting the graph up or down does not limit the values \(f(x)\) can take. The output still extends from \( -\infty \) to \( \infty \).

### Conclusion:
The range of \( f(x) = \sqrt[3]{x+6} - 2 \) in interval notation is \( (-\infty, \infty) \), which corresponds to **option D**.

### Answer: **D. (−∞, ∞)**

Do you want further details on transformations or any other questions? Here are 5 related questions:

1. How do vertical shifts affect the range of functions?
2. What is the domain of a cube root function?
3. How does the graph of the cube root function compare to the square root function?
4. How do horizontal translations affect the domain of functions?
5. What is the general form of a translated function and how to interpret it?

**Tip:** When analyzing translations of functions, always start by identifying how horizontal and vertical shifts affect both the domain and range.

The graph of the cube root parent function was translated to create the graph f(x)=(x+6)−2, which is shown in the graph below. Which of the following represents the range of f(x) in interval notation?
A.(−4, 1)
B.(−6, ∞)
C.[−4, 1]
D.(−∞, ∞)

Explore how the cube root function f(x)=(x+6)-2 is translated and learn how to determine its range in interval notation. This problem includes algebraic transformations, cube root functions, and range analysis. Suitable for high school students in Grades 9-12.

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