To determine the domain and range of the function [tex]\( f(x) = 4 \left(\sqrt[3]{81}\right)^x \)[/tex], let's analyze the function step-by-step.
1. Simplifying the base:
- The base inside the function is [tex]\( \sqrt[3]{81} \)[/tex]. The cube root of 81 is approximately 4.3267487109222245.
2. Domain:
- The domain of an exponential function of the form [tex]\( a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants and [tex]\( b > 0 \)[/tex], includes all real numbers. There are no restrictions on the values that [tex]\( x \)[/tex] can take.
- Therefore, the domain of [tex]\( f(x) = 4 \left(4.3267487109222245\right)^x \)[/tex] is all real numbers.
3. Range:
- The range of an exponential function of the form [tex]\( a \cdot b^x \)[/tex], where [tex]\( a > 0 \)[/tex] and [tex]\( b > 0 \)[/tex], is [tex]\( y > 0 \)[/tex]. This is because the exponential function [tex]\( b^x \)[/tex] for [tex]\( b > 1 \)[/tex] continuously outputs positive values for any real number exponent [tex]\( x \)[/tex], and multiplying by a positive constant [tex]\( a \)[/tex] retains this property.
- Therefore, the range of [tex]\( f(x) = 4 \left(4.3267487109222245\right)^x \)[/tex] is [tex]\( y > 0 \)[/tex].
Given these analyses, the correct answer is:
[tex]\[
\{ x \mid x \text{ is a real number} \} ; \{ y \mid y > 0 \}
\][/tex]
