To determine the domain and range of the function [tex]\( f(x) = 4 (\sqrt[3]{81})^x \)[/tex], let's take a step-by-step approach.
### Step 1: Understanding the Base
First, simplify the base [tex]\( \sqrt[3]{81} \)[/tex]:
[tex]\[ \sqrt[3]{81} = 81^{1/3} \][/tex]
Given that [tex]\( 81 = 3^4 \)[/tex], the expression becomes:
[tex]\[ (\sqrt[3]{81}) = (3^4)^{1/3} = 3^{4/3} \][/tex]
So, the function can be rewritten as:
[tex]\[ f(x) = 4 \cdot (3^{4/3})^x \][/tex]
### Step 2: Simplified Form of the Function
When simplifying further, the exponent rules tell us:
[tex]\[ (a^m)^n = a^{m \cdot n} \][/tex]
Thus:
[tex]\[ f(x) = 4 \cdot 3^{(4/3)x} \][/tex]
### Step 3: Domain
The domain of a function is the set of all possible input values (x-values) that the function accepts. For the function:
[tex]\[ f(x) = 4 \cdot 3^{(4/3)x} \][/tex]
Since there are no restrictions on [tex]\( x \)[/tex] (such as division by zero or square roots of negative numbers), the function is defined for all real numbers. Thus, the domain is:
[tex]\[ \{x \mid x \text{ is a real number}\} \][/tex]
### Step 4: Range
The range of a function is the set of all possible output values (y-values). To analyze this, notice:
- The expression [tex]\( 3^{(4/3)x} \)[/tex] is an exponential expression where the base [tex]\( 3^{4/3} \)[/tex] is a positive number greater than 1.
- Exponential functions of the form [tex]\( b^x \)[/tex] where [tex]\( b > 1 \)[/tex] always yield positive output values for all real [tex]\( x \)[/tex] and the values can get arbitrarily large.
- The coefficient "4" in [tex]\( 4 \cdot 3^{(4/3)x} \)[/tex] scales the values of [tex]\( 3^{(4/3)x} \)[/tex] but does not affect the positivity.
Since [tex]\( 3^{(4/3)x} \)[/tex] is always positive, [tex]\( 4 \cdot 3^{(4/3)x} \)[/tex] must also always be positive. Therefore, the range is:
[tex]\[ \{ y \mid y > 0 \} \][/tex]
### Conclusion
Given the analysis, the domain and range of the function [tex]\( f(x) = 4(\sqrt[3]{81})^x \)[/tex] are:
- Domain: [tex]\(\{ x \mid x \text{ is a real number} \}\)[/tex]
- Range: [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]
