The options that are equivalent to the given function are;
Option A
Option B
Option E
The correct options are;
A) f(x) = [tex]162^{x/4}[/tex]
B) f(x) = (3∜2[tex])^{x}[/tex]
C) f(x) = [tex]9\sqrt[4]{2^{x} }[/tex]
D) f(x) = [tex]162^{4/x}[/tex]
E) f(x) = [tex](3(2^{1/4}))^{x}[/tex]
We are given the function; f(x) = [tex](\sqrt[4]{162})^{x}[/tex]
Now, when we have something like; [tex]\sqrt[b]{2}^{a}[/tex]
This expression can also be expressed as; [tex]2^{a/b}[/tex]
Therefore our function can also be expressed as; f(x) = [tex]162^{x/4}[/tex]
Option A; This is correct because it is equal to our function gotten
Option B; f(x) = (3∜2[tex])^{x}[/tex]
From earlier, we will have; (3 × [tex]2^{1/4}[/tex][tex])^{x}[/tex]
Now, 3 can also be expressed as [tex]81^{1/4}[/tex]
Thus, we now have;
([tex]81^{1/4}[/tex] × [tex]2^{1/4}[/tex][tex])^{x}[/tex]
Factorizing out the 1/4 exponent gives;
(81 × 2[tex])^{x/4}[/tex]
⇒ = [tex]162^{x/4}[/tex]
This is same with our function and it is correct
Option C; f(x) = (9∜2[tex])^{x}[/tex]
This is similar to our equation in option B which was correct but the difference here is 9 instead of 3 and clearly this option is not correct.
Option D; f(x) = [tex]162^{4/x}[/tex]
This is not the same with our function and so it is wrong
Option E; f(x) = [tex](3(2^{1/4}))^{x}[/tex]
As seen in option B;
3 can also be expressed as [tex]81^{1/4}[/tex]
Thus, we now have;
;([tex]81^{1/4}[/tex] × [tex]2^{1/4}[/tex][tex])^{x}[/tex]
Factorizing out the 1/4 exponent gives;
(81 × 2[tex])^{x/4}[/tex]
⇒ = [tex]162^{x/4}[/tex]
This is same with our function and it is correct
Read more at; https://brainly.com/question/6504126
