Which functions are equivalent to [tex]f(x)=\sqrt[4]{162}^x[/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 base of the original function: 4 162 = 3 4 2 .
Rewrite the original function: f ( x ) = ( 3 4 2 ) x .
Check each option for equivalence by comparing it to the simplified form.
The equivalent functions are: f ( x ) = 16 2 4 x , f ( x ) = ( 3 4 2 ) x , and f ( x ) = [ 3 ( 2 4 1 ) ] x .
Explanation
Problem Analysis We are given the function f ( x ) = 4 162 x and asked to identify equivalent functions from a list of options. Our strategy will be to simplify the given function and compare it to the options.
Simplifying the Base First, let's simplify the base of the original function, 4 162 . We can write 162 as 2 ⋅ 81 = 2 ⋅ 3 4 . Therefore, we have 4 162 = 4 2 ⋅ 3 4 = 4 2 ⋅ 4 3 4 = 3 4 2 So, f ( x ) = ( 3 4 2 ) x .
Checking Each Option Now, let's examine each option to see if it is equivalent to f ( x ) = ( 3 4 2 ) x .
Option 1: f ( x ) = 16 2 4 x We can rewrite this as f ( x ) = ( 16 2 4 1 ) x = ( 4 162 ) x = ( 3 4 2 ) x . So, this option is equivalent.
Option 2: f ( x ) = ( 3 4 2 ) x This is the simplified form of the original function, so it is equivalent.
Option 3: f ( x ) = 9 4 2 x This can be written as f ( x ) = 9 ( 2 4 1 ) x . Since 9 = 3 , this is not equivalent.
Option 4: f ( x ) = 16 2 x 4 This is not equivalent since the exponent is x 4 instead of x .
Option 5: f ( x ) = [ 3 ( 2 4 1 ) ] x This is the same as f ( x ) = ( 3 4 2 ) x , so it is equivalent.
Final Answer Therefore, the functions equivalent to f ( x ) = 4 162 x are:
f ( x ) = 16 2 4 x f ( x ) = ( 3 4 2 ) x f ( x ) = [ 3 ( 2 4 1 ) ] x
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population of bacteria doubles every hour, the population size can be modeled by an exponential function. Similarly, the decay of a radioactive substance can be modeled using an exponential function with a negative exponent. Understanding how to manipulate and simplify exponential functions is crucial in these applications.
The equivalent functions to f ( x ) = 4 162 x are f ( x ) = 16 2 4 x , f ( x ) = ( 3 4 2 ) x , and f ( x ) = [ 3 ( 2 4 1 ) ] x . The non-equivalent functions are f ( x ) = 9 4 2 x and f ( x ) = 16 2 x 4 .
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