Problem 58 - A Complete Resource Book in Mathematics for JEE Main

Question

The function \(f(x)\) is defined in \([0,1]\), then the domain of definition of the function \(f\left[\log \left(1-x^{2}\right)\right]\), is (A) \(x \in\\{0\\}\) (B) \(x \in[-\sqrt{1+e},-1] \cup[1,+\sqrt{1+e}]\) (C) \(x \in(-\infty, \infty)\) (D) None of these

Step-by-Step Solution

Answer

(D) None of these

Step 1: Understand the Inside Function

Consider the function inside the brackets: \( \log(1-x^2) \). This function is defined only when its argument is positive, i.e., \(1-x^2 > 0\). We need to determine the values of \(x\) for which this inequality holds true.

Step 2: Solve the Inequality

Solve the inequality \(1-x^2 > 0\). This simplifies to \(x^2 < 1\). Solving this inequality for \(x\) gives \(|x| < 1\). Hence, \(x\) must belong to the open interval \((-1, 1)\).

Step 3: Relate to Function Domain

We're given that \(f(x)\) is defined on \([0, 1]\). Therefore, the domain of \(f[\log(1-x^2)]\) will be the intersection of \(x \in (0, 1)\) coming from step 2 and \([0, 1]\) coming from the definition of \(f(x)\).

Step 4: Find the Domain of Function

The intersection of \((0, 1)\) with \([0, 1]\) is \((0, 1)\). Thus, the valid domain for the entire function \(f[\log(1-x^2)]\) is \(x \in (0, 1)\).

Step 5: Verify With Options

Review the given options. None of the options specifically list \(x \in (0, 1)\). Hence, none of the provided options are correct.

Key Concepts

Logarithmic Function

Logarithmic functions are an essential part of understanding many mathematical concepts. They are the inverse of exponential functions, meaning they "undo" exponential growth. If we say that a number is expressed as an exponent of a base, logarithms determine the power to which the base is raised to yield that number. For instance, in a logarithmic function like \( \log_b(x) \), \( b \) is the base, and we find the power needed to raise \( b \) to obtain \( x \).

Understanding Logarithmic Functions:

• The domain of a logarithmic function \( \log_b(x) \) is all positive numbers because you can only take the logarithm of positive values.
• The base, \( b \), must be a positive number and cannot equal 1.
• A logarithmic function like \( \log(1-x^2) \) requires that \( 1-x^2 > 0 \) because the input must be greater than zero for the log function to be defined.

Understanding the basic rules and conditions of logarithmic functions helps solve many exercises, like determining the domain of such functions.

Inequality

In mathematics, inequalities are used to compare expressions or numbers. These comparisons allow us to find ranges of values for which certain expressions are valid. Inequalities use symbols such as "<" or ">" to describe the relationship between values.

Key Points about Inequalities:

• Inequalities can be linear or non-linear. A linear inequality is a first-degree relation between variables, as in \( 1 - x^2 > 0 \).
• Solving a quadratic inequality like \( 1 - x^2 > 0 \) involves finding when the expression is greater than zero. This typically requires finding the roots and testing intervals between and beyond these roots.
• For the inequality \( 1 - x^2 > 0 \), solving it gives \( |x| < 1 \), meaning the range of values for \( x \) lies between \(-1\) and \(1\).
By comprehending inequalities, you can set the bounds for functions and evaluate which values make them valid.

Interval Notation

Interval notation is a convenient and compact way to describe a set of numbers that lie between two endpoints on the number line. It's especially useful for representing solutions to inequalities and the domains of functions.

Understanding Interval Notation:

• Intervals can be open \((a, b)\), closed \([a, b]\), or half-open/half-closed \((a, b]\) or \([a, b)\). The parentheses \(()\) indicate that the endpoints are not included, while square brackets \([]\) mean the endpoints are included.
• For example, the interval \((0, 1)\) indicates all real numbers between 0 and 1 but does not include 0 and 1 themselves.
• In the context of the exercise, the domain for \( f[\log(1-x^2)] \) after considering the intersection of constraints was \((0, 1)\), highlighting that \( x \) must be strictly greater than 0 and less than 1.

Using interval notation simplifies the process of writing and analyzing the solutions of inequalities or domain constraints, making mathematical communication clearer.