Let f(x)=√(a-x^3+3 b x^2-3 b x+b^3+b) . Find b , if f(x) is inverse of itself.

Name: Problem 124, Chapter 1: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-04-15T16:42:40-08:00
Duration: 1 min 51 s
Channel: Shima Shaw
Description: Let f(x)=√(a-x^3+3 b x^2-3 b x+b^3+b) . Find b , if f(x) is inverse of itself.

step 1 Hello everyone. From the question we are given that fx is equal to cube root of a minus x to the

Let f(x)=√(a-x^3+3 b x^2-3 b x+b^3+b) . Find b , ...

Key Concepts

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a given function f, its inverse, denoted f?¹, satisfies the condition that f(f?¹(x)) = x for all x in the domain. This concept is central when trying to 'undo' the actions of a function, and finding the inverse involves isolating the variable so that the roles of input and output are swapped.

Self-Inverse Functions

A self-inverse function is one in which the function is its own inverse; that is, f(x) = f?¹(x) for every x in the domain. This means that if you apply the function twice, you retrieve the original input: f(f(x)) = x. This property typically imposes symmetry about the line y = x, and determining conditions for this property is often key in problems involving function inverses.

Function Composition

Function composition involves applying one function to the result of another, denoted as (f ? g)(x) = f(g(x)). In the context of inverse functions, the composition of a function and its inverse results in the identity function. Verifying that f(f(x)) = x is a method used to ensure that the function is self-inverse.

Polynomial Transformation

In many problems, especially those involving functions defined by polynomials, recognizing and manipulating polynomial expressions can be vital. This might involve factoring or rearranging a polynomial to reveal structural symmetries or to simplify the expression. Such transformations help in aligning terms and coefficients to satisfy conditions for properties like self-inverseness.

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