Simplify the function f(x)=1/3(81)^3x/4 . Then determine the key aspects of the function.

Answer:[tex]f(x)=3^{3x-1}[/tex]. The domain of the function is the set of all real number and the range is [tex](0,\infty)[/tex]Step-by-step explanation:Given:The function is  given as:[tex]f(x)=\frac{1}{3}(81)^{\frac{3x}{4}}[/tex]Using the rule of the exponents, [tex]a^{mn}=(a^m)^n[/tex],[tex]f(x)=\frac{1}{3}((81)^{\frac{1}{4}})^{(3x)}\\f(x)=\frac{1}{3}(\sqrt[4]{81} )^{3x}\\f(x)=\frac{1}{3}(3)^{3x}\\f(x)=\frac{3^{3x}}{3^1}[/tex]Using the rule of the exponents,[tex]\frac{a^m}{a^n}=a^{m-n}[/tex],[tex]f(x)=3^{3x-1}[/tex]Therefore, the simplified form of the given function is:[tex]f(x)=3^{3x-1}[/tex]Key aspects:The given function is an exponential function with a constant base 3.Domain is the set of all possible values of [tex]x[/tex] for which the function is defined.The domain of an exponential function is a set of all real values.The range of an exponential function is always greater than zero.Therefore, the domain of this function is also all real values and the range is from 0 to infinity.Domain: [tex]x \epsilon (-\infty,\infty)[/tex]Range: [tex]y\epsilon (0,\infty)[/tex]