Which functions are equivalent to [tex]f(x)=\sqrt[4]{162} \times[/tex] ? Check all that apply.
[tex]f(x)=162^{\frac{x}{4}}[/tex]
[tex]f(x)=(3 \sqrt[4]{2})^x[/tex]
[tex]f(x)=9 \sqrt[4]{2}^x[/tex]
[tex]f(x)=162^{\frac{4}{x}}[/tex]
[tex]f(x)=\left[3\left(2^{\frac{1}{4}}\right)\right]^x[/tex]
Answer (2)
Simplify the original function: f ( x ) = 4 162 x = 3 4 2 x .
Analyze each option and rewrite them in terms of exponents.
Compare each option with the simplified original function.
Determine that none of the options are equivalent to the original function because the original function is linear while the options are exponential. N o n e
Explanation
Problem Analysis We are given the function f ( x ) = 4 162 x and asked to determine which of the following functions are equivalent to it:
f ( x ) = 16 2 4 x
f ( x ) = ( 3 4 2 ) x
f ( x ) = 9 4 2 x
f ( x ) = 16 2 x 4
f ( x ) = [ 3 ( 2 4 1 ) ] x
To solve this, we need to simplify the original function and compare it to the options.
Simplifying the Original Function First, let's simplify f ( x ) = 4 162 x . We can rewrite 162 as 81 × 2 = 3 4 × 2 . Therefore,
4 162 = 4 3 4 × 2 = 4 3 4 × 4 2 = 3 4 2 .
So, f ( x ) = 3 4 2 x .
Analyzing Each Option Now, let's analyze each option:
f ( x ) = 16 2 4 x = ( 3 4 × 2 ) 4 x = 3 4 × 4 x × 2 4 x = 3 x 2 4 x . This is not equivalent to 3 4 2 x .
f ( x ) = ( 3 4 2 ) x = ( 3 × 2 4 1 ) x = 3 x × 2 4 x . This is not equivalent to 3 4 2 x .
f ( x ) = 9 4 2 x = 9 × ( 2 4 1 ) x = 3 2 × 2 4 x . This is not equivalent to 3 4 2 x .
f ( x ) = 16 2 x 4 = ( 3 4 × 2 ) x 4 = 3 x 16 × 2 x 4 . This is not equivalent to 3 4 2 x .
f ( x ) = [ 3 ( 2 4 1 ) ] x = ( 3 × 2 4 1 ) x = 3 x × 2 4 x . This is not equivalent to 3 4 2 x .
Final Comparison None of the given functions are equivalent to f ( x ) = 3 4 2 x . The key difference is that the original function is a linear function of x , while all the options are exponential functions of x .
Conclusion Therefore, none of the provided functions are equivalent to the original function.
Examples
In signal processing, understanding the equivalence of functions is crucial when analyzing and manipulating signals. For example, if you have a signal represented by f ( x ) = 4 162 x , and you need to process it using a system that only accepts exponential functions, you would need to find an appropriate approximation or transformation. Knowing that none of the exponential functions provided are equivalent helps you avoid incorrect substitutions and ensures the integrity of the signal processing pipeline. This is also applicable in machine learning when choosing appropriate activation functions or feature transformations.
None of the given functions are equivalent to f ( x ) = 4 162 ⋅ x . The original function simplifies to a linear form, whereas all options presented are exponential in nature. Therefore, the conclusion is that none of the listed functions match the original function.
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