Let's determine both the domain and range of the function [tex]\( f(x) = 4 (\sqrt[3]{81})^x \)[/tex].
### Finding the Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
For the function [tex]\( f(x) = 4 (\sqrt[3]{81})^x \)[/tex]:
- The base, [tex]\(\sqrt[3]{81}\)[/tex], is a positive real number. The cube root of 81 is approximately [tex]\(4.3267\)[/tex], but it's sufficient to understand that the function involves raising a constant base to any real number power [tex]\(x\)[/tex].
- Exponential functions with a real constant base (not equal to zero or negative) raised to the power [tex]\(x\)[/tex] are defined for all real numbers [tex]\(x\)[/tex].
Therefore, the domain of the function [tex]\( f(x) = 4 (\sqrt[3]{81})^x \)[/tex] is:
[tex]\[ \{x \mid x \text{ is a real number}\} \][/tex]
### Finding the Range:
The range of a function is the set of all possible output values (y-values) that the function can take.
Consider the function [tex]\( f(x) = 4 (\sqrt[3]{81})^x \)[/tex]:
- Since [tex]\(\sqrt[3]{81}\)[/tex] is approximately equal to 4.3267 (and it's positive and greater than 1), raising it to any power [tex]\( x \)[/tex] will always be a positive number.
- Multiplying this positive number by 4 (which is also positive) ensures that the result is a positive number.
Therefore, the function [tex]\( f(x) = 4 (\sqrt[3]{81})^x \)[/tex] will only produce positive output values, regardless of the value of [tex]\( x \)[/tex].
Thus, the range of the function is:
[tex]\[ \{y \mid y > 0\} \][/tex]
### Conclusion:
Combining the domain and range:
- The domain of the function is [tex]\(\{x \mid x \text{ is a real number}\}\)[/tex].
- The range of the function is [tex]\(\{y \mid y > 0\}\)[/tex].
So, the correct answer is:
[tex]\[ \{x \mid x \text{ is a real number}\} ; \{y \mid y > 0\} \][/tex]
