Find the domains of f(x)=cos^-1((x)/(2)-1)+e^x+(1)/(⌊x-1 |)...

Name: Problem 127, Chapter 1: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-02-26T15:06:36-08:00
Duration: 1 min 37 s
Channel: Varsha Aggarwal
Description: Find the domains of f(x)=cos^-1((x)/(2)-1)+e^x+(1)/(⌊x-1 |)

Key Concepts

Exponential Function Domain

The exponential function, such as e^x, is defined for all real numbers. Hence, it does not impose any restrictions on the domain of the composite function. However, it must still be considered along with other parts of the expression to determine the overall domain.

Absolute Value Function

The absolute value function transforms any real number to its non?negative counterpart. When used inside other functions, such as the floor function, it limits the possible outputs and affects the interval calculations. Understanding the behavior of the absolute value function is essential for determining which values of the variable will cause issues, like creating a zero in the denominator when combined with the floor function.

Inverse Cosine Domain Restrictions

The arccos or inverse cosine function is only defined for input values between -1 and 1. In composite functions, any expression placed inside the arccos must satisfy this inequality. This concept ensures that when solving for the domain of a function containing an inverse trigonometric component, one must restrict the variable such that the inner expression stays within [-1, 1].

Floor Function and Division by Zero

The floor function takes a real number and returns the greatest integer less than or equal to that number. When the floor function appears in the denominator, it is imperative to ensure that it does not evaluate to zero because division by zero is undefined. This concept requires analyzing the range of the expression within the floor function and identifying the regions where it might produce a zero to exclude those from the domain.

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