Sure, let's explore this problem step-by-step.
1. Understanding the Parent Function [tex]\(f(x) = \sqrt[3]{x}\)[/tex]:
- Domain: The function [tex]\(f(x) = \sqrt[3]{x}\)[/tex] represents the cube root of [tex]\(x\)[/tex]. Since you can take the cube root of any real number (both positive and negative), the domain of [tex]\(f(x)\)[/tex] is all real numbers.
- Range: Similarly, the output of [tex]\(f(x) = \sqrt[3]{x}\)[/tex] can also be any real number because cube roots can produce any real value. Thus, the range of [tex]\(f(x)\)[/tex] is all real numbers.
2. Examining the Function [tex]\(g(x) = \sqrt[3]{x+6} - 8\)[/tex]:
- Domain: The expression [tex]\(x + 6\)[/tex] inside the cube root can take any real number because you can add 6 to any real number [tex]\(x\)[/tex]. Hence, the domain of [tex]\(g(x)\)[/tex] is also all real numbers.
- Range: Regarding the range, we should note that [tex]\(g(x)\)[/tex] is essentially a transformation of the parent function [tex]\(f(x)\)[/tex]. It involves a horizontal shift by -6 (i.e., [tex]\(x + 6\)[/tex]) and a vertical shift by -8 (i.e., [tex]\(\sqrt[3]{x+6} - 8\)[/tex]). However, these shifts do not restrict the possible values of the function significantly. Since any real number can be shifted horizontally and vertically without restriction, the cube root part still covers all real numbers, and the addition/subtraction shifts the entire range accordingly. Thus, the range remains all real numbers.
3. Conclusion:
Both [tex]\(g(x) = \sqrt[3]{x+6} - 8\)[/tex] and [tex]\(f(x) = \sqrt[3]{x}\)[/tex] have the same domain and range, which are all real numbers.
Therefore, the statement that best describes [tex]\(g(x)\)[/tex] and the parent function [tex]\(f(x)\)[/tex] is:
"The ranges of [tex]\(g(x)\)[/tex] and [tex]\(f(x)\)[/tex] are the same, and their domains are also the same."
This matches the third option provided:
```
The ranges of [tex]\(g(x)\)[/tex] and [tex]\(f(x)\)[/tex] are the same, and their domains are also the same.
```
