Let f(x) = (x^x)^x and g(x) = x^{(x^x)}. Determine the relationship between f(x) and g(x).

The function g ( x ) = x ( x x ) grows faster than f ( x ) = ( x x ) x for all positive values of x . This is due to the fact that the exponent in g ( x ) , which is x x , increases more rapidly than the exponent in f ( x ) , which is x 2 . Therefore, we conclude that f(x)"> g ( x ) > f ( x ) for 0"> x > 0 .
;

Answered by Anonymous | 2025-07-04

To determine the relationship between f ( x ) = ( x x ) x and g ( x ) = x ( x x ) , let's analyze their structures and simplify them as much as possible.

Understanding f ( x ) = ( x x ) x :


The expression f ( x ) = ( x x ) x indicates that you first raise x to the power of x , which gives x x , and then raise the result to the power of x again. This results in: f ( x ) = ( x x ) x = x ( x ⋅ x ) = x x 2


Understanding g ( x ) = x ( x x ) :


The function g ( x ) = x ( x x ) involves raising x to a power of x x . So it remains: g ( x ) = x ( x x )


Comparing f ( x ) and g ( x ) :


We now have f ( x ) = x x 2 and g ( x ) = x ( x x ) .

The key difference here is in the exponents:

For f ( x ) , the exponent is x 2 , which means x im es x .

For g ( x ) , the exponent is x x , where x is raised to the power of itself.



Conclusion:


Although both functions involve exponential operations with repeated bases and powers, they aren't equivalent due to their differing exponents. While g ( x ) represents iterating the exponent with x self-multiply, f ( x ) involves simply squaring the base exponent.

Therefore, f ( x ) and g ( x ) represent different mathematical expressions and are not equal to each other. They grow at different rates due to their differing exponents, with g ( x ) potentially growing faster depending on the value of x .

Answered by ElijahBenjaminCarter | 2025-07-06