Problem 22 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

Domain of definition of the function \(f(x)=\frac{3}{4-x^{2}}+\log _{10}\left(x^{3}-x\right)\) is (a) \((-1,0) \cup(1,2) \cup(2, \infty)\) (b) \((1,2)\) (c) \((-1,0) \cup(1,2)\) (d) \((1,2) \cup(2, \infty)\)

Step-by-Step Solution

Answer

The domain is \((-\infty, -1) \cup (1, 2) \cup (2, \infty)\), which corresponds to option (d): \((1, 2) \cup (2, \infty)\).

Step 1: Determine restrictions from the fraction

The function has a fraction \( \frac{3}{4-x^2} \). We know a fraction is undefined if its denominator is zero. \[ 4 - x^2 = 0 \] Solve for \( x \): \[ x^2 = 4 \] \[ x = \pm 2 \] Thus, the function is undefined at \( x = 2 \text{ and } x = -2 \).

Step 2: Determine restrictions from the logarithm domain

The function also includes a logarithm, \( \log_{10}(x^3 - x) \). The argument of a logarithm must be greater than zero:\[ x^3 - x > 0 \] Factor out \( x \): \[ x(x^2 - 1) > 0 \] Further factor \( x^2 - 1 \): \[ x(x - 1)(x + 1) > 0 \] Identify intervals by analyzing critical points: \( x = -1, 0, 1 \). Evaluate sign changes across intervals.

Step 3: Analyze sign changes and intervals

Evaluate sign of the inequality \( x(x - 1)(x + 1) > 0 \) over the intervals determined:- For \((-\infty, -1)\), the inequality is positive.- For \((-1, 0)\), the inequality is negative.- For \((0, 1)\), the inequality is negative.- For \((1, \infty)\), the inequality is positive.Thus, the intervals where the inequality holds and \( \log_{10}(x^3-x) \) is defined are \((-\infty, -1)\) and \((1, \infty)\).

Step 4: Combine results

Combine the results from both the denominator and logarithm to find allowable \( x \): - From the fraction: \( x eq \pm 2 \)- From the logarithm: \( x \in (-\infty, -1) \cup (1, \infty) \)Thus, \( f(x) \) is defined for \( x \in (-\infty, -1) \cup (1, 2) \cup (2, \infty) \).

Key Concepts

Logarithmic Functions

Understanding logarithmic functions is crucial for solving problems involving their domain. A logarithm, written as \( \log_{10}(x) \), is essentially the inverse of an exponent for a base of 10. It answers the question: "To what exponent must 10 be raised, to yield \( x \)?"

In any logarithmic expression, the argument \( x \) (inside the log) must be greater than zero. This is a vital rule because a logarithm of a negative value or zero is not defined within real numbers.

For instance, when dealing with the given logarithmic component \( \log_{10}(x^3-x) \), we set up the inequality \( x^3 - x > 0 \). Solving this inequality helps us find the intervals where the log function is defined, which is crucial for finding the domain of the overall function.

Inequalities

Inequalities are expressions that compare two values, often using symbols like \( > \), \( < \), \( \geq \), or \( \leq \). Solving inequalities involves finding a range or set of values that satisfy the condition.

In our exercise, we deal with the inequality \( x(x - 1)(x + 1) > 0 \). A common technique to solve such inequalities is interval analysis. This involves identifying critical points where the expression equals zero, then testing sign changes across the resulting intervals.

Here, critical points are \( x = -1, 0, 1 \). These points split the number line into intervals like \( (-\infty, -1) \), \( (-1, 0) \), \( (0, 1) \), and \( (1, \infty) \). By choosing test points from these intervals, we determine where the inequality holds true.

Understanding how to analyze and solve inequalities is essential not only in this exercise but also in many mathematical applications involving functions and their domains.

Rational Functions

Rational functions are expressions where one polynomial is divided by another. A simple example might be \( \frac{3}{4-x^2} \). Similar to other fractions, a rational function becomes undefined when the denominator is zero. Therefore, finding the domain of a rational function involves determining where the denominator does not equal zero.

In the provided exercise, by setting the denominator \( 4 - x^2 = 0 \), we solve for \( x \) and find \( x = \pm 2 \). This tells us that the rational function is undefined at these points, hence \( x = 2 \) and \( x = -2 \) must be excluded from the domain.

Understanding rational functions is important because it helps you unearth the structure of a function's graph and its potential asymptotes or discontinuities. Recognizing these will aid in analyzing and understanding complex functions in both academic and real-world scenarios.