Find the domain of f(x)=sin^-1((1+x^2)/(2 x))+cos(sinx) |...

Name: Problem 131, Chapter 1: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-02-26T15:20:40-08:00
Duration: 1 min 42 s
Channel: Varsha Aggarwal
Description: Find the domain of f(x)=sin^-1((1+x^2)/(2 x))+cos(sinx)

Key Concepts

Inverse Sine Function Domain

The inverse sine function, commonly denoted as arcsin, is defined only when its argument is between -1 and 1. This restriction is crucial since any expression placed inside the arcsin must evaluate to a number within this closed interval for the function to exist. Understanding and applying this condition is essential when determining the domain of any function that includes arcsin.

Domain of Composite Functions

When a function is composed of multiple parts, the overall domain is determined by the intersection of the domains of the individual components. In this context, one must ensure that each subfunction receives inputs for which it is defined. This systematic approach is particularly important when working with both trigonometric and inverse trigonometric functions together.

Rational Inequalities

Rational inequalities involve expressions where variables appear in the denominator, and solving them generally requires careful analysis based on the sign of the variable. Multiplying an inequality by an expression containing the variable can reverse the direction of the inequality if the expression is negative, so it is essential to consider different cases (such as when the variable is positive or negative) to correctly identify the solution set.

Trigonometric Function Properties

Basic trigonometric functions, like sine and cosine, are defined for all real numbers. For instance, a composition such as cos(sin x) does not impose additional restrictions on the domain because both the sine and cosine functions are universally defined over real numbers. This property often simplifies the analysis of the overall function's domain, allowing focus on the more restrictive components.

Transcript

Please give Ace some feedback