Answer:
The simplified form of the given function is [tex]f(x)=3^{3x-1}[/tex]. The domain of the functuon is the set of all real number and the range is [tex](0,\infty)[/tex].
Step-by-step explanation:
The given function is
[tex]f(x)=\frac{1}{3}(81)^{\frac{3x}{4}[/tex]
Use the exponent rule [tex]a^{mn}=(a^m)^n[/tex],
[tex]f(x)=\frac{1}{3}((81)^{\frac{1}{4})^{3x}[/tex]
[tex]f(x)=\frac{1}{3}(3)^{3x}[/tex]
Use the exponent rule [tex]\frac{a^m}{a^m}=a^{m-n}[/tex],
[tex]f(x)=3^{3x-1}[/tex]
The simplified form of the given function is,
[tex]f(x)=3^{3x-1}[/tex]
Domain is the set of all possible values of x for which the function is defined.
Since the function is defined for all real values of x, therefore domain is the set of all real.
The value of function is always positive for any value of x. The value of a function is a large number when x approaches too infinity. The value of the function approaches to 0 as x approaches to negative infinity.
Therefore the range of the function is always greater than 0.
