Given a function y=x^x, x>0 and 0<x<1 . The values of x for...

Given a function $y=x^{x}, x>0$ and $0<x<1 .$ The values of $x$ for which the function attain values exceeding the values of its inverse are
(a) $0<x<1$
(b) $1<x<\infty$
(c) $0<x<2$
(d) None of these

Key Concepts

Exponential-type Functions and Their Behavior

Functions involving exponents, like x raised to the power x, exhibit distinctive types of growth or decay based on the input value. Understanding the behavior of these functions, including how they increase or decrease and how they compare to their inverses, is key to determining intervals where one might exceed the other.

Inverse Functions

An inverse function essentially undoes the action of the original function, meaning that if f is the original function, then f?¹ is defined such that f(f?¹(x)) = x and f?¹(f(x)) = x. This concept is critical when comparing a function with its inverse, as it allows exploring symmetry and relationships between the two functions.

Fixed Points and Intersection with the Line y = x

Fixed points occur where a function's value equals its input (f(x) = x). For any function and its inverse, these fixed points are significant because they are exactly the points at which the function and its inverse coincide, serving as boundary markers for analyzing regions where one may be greater than the other.

Monotonicity

The property of monotonicity, where a function is either entirely increasing or decreasing, ensures that the function has an inverse and that the relationship between the function and its inverse can be examined on a well-defined interval. When a function is monotonic, comparisons between the function and its inverse over certain intervals become simpler to analyze.

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