Let g(x)=1+x-[x] and f(x)={-1 : x 0 . then for all x, find f(g(x))

Name: Problem 149, Chapter 1: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-02-26T17:49:14-08:00
Duration: 1 min 29 s
Channel: Varsha Aggarwal
Description: Let g(x)=1+x-[x] and f(x)={-1 : x 0 . then for all x, find f(g(x))

step 1 So f g x x x equal f 1 plus x minus greater integer of x so so range of range of g is this funct

Let g(x)=1+x-[x] and f(x)={-1 : x<0 0 : x=0 1 :...

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, as in f(g(x)). It is an important technique in mathematics for building complex functions from simpler ones and understanding how the properties of the individual functions combine. In this problem, composing the functions allows the transformation of g(x) through f.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input domain. They are essential when a function’s behavior changes across different intervals. In the given problem, f(x) is defined piecewise based on whether the input is negative, zero, or positive, which is a common scenario in function analysis and modeling.

Floor Function

The floor function, denoted by [x], returns the greatest integer that is less than or equal to x. It is a fundamental concept in real analysis and number theory, used to separate the integral and fractional parts of a real number. This function is key in problems where discretization or rounding down plays a role.

Fractional Part Function

The fractional part function is defined as the difference between a real number and its floor, that is, {x} = x - [x]. It captures the 'remainder' when a number is divided into its integer and non-integer components. This concept is critical in transforming expressions involving x and its floor function into a more manageable form.

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