Problem 18 - Calculus for JEE ADVANCED

Question

Here, \(x+3>0\) and \(x^{2}+3 x+2 \neq 0\) Therefore, \(x>-3\) and \((x+1)(x+2) \neq 0\), i.e., \(x \neq-1,-2\). Therefore, the domain is \((-3, \infty)-\\{-1,-2\\}\).

Step-by-Step Solution

Answer

The domain is \((-3, \infty) \setminus \{-1, -2\}\).

Step 1: Analyze Inequality

First, analyze the inequality \(x + 3 > 0\). Solve for \(x\) by subtracting 3 from both sides of the inequality, which gives \(x > -3\). This indicates that \(x\) must be greater than -3.

Step 2: Analyze the Quadratic Constraint

Next, address the inequality from the second constraint, \(x^2 + 3x + 2 eq 0\). Factor the quadratic expression to get \((x + 1)(x + 2) eq 0\). This implies that \(x\) cannot be equal to -1 or -2.

Step 3: Determine the Domain

Combine the results from Step 1 and Step 2. Since \(x > -3\) and \(x eq -1, -2\), the domain excludes -1 and -2 from the interval greater than -3. Therefore, the domain is expressed as \((-3, \infty) \setminus \{-1, -2\}\).

Key Concepts

Inequalities

Inequalities are statements that compare two values to show if one is less than, greater than, or simply not equal to another value. In the given problem, we begin with the inequality \(x + 3 > 0\). To find the solution, we need to isolate \(x\). This is done by subtracting 3 from both sides, which results in \(x > -3\). This inequality shows that \(x\) can be any real number greater than \(-3\).

Understanding inequalities is crucial when determining the set of values (domain) for which a function is defined. It's important to remember that when we multiply or divide both sides of an inequality by a negative number, the inequality sign flips. This isn't the case here but is a good rule to keep in mind for future problems. Always check your final solution by plugging in values to ensure it satisfies the original inequality.

Quadratic Equations

Quadratic equations are polynomial equations of degree 2. They are often written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In this exercise, the quadratic equation is given as \(x^2 + 3x + 2 eq 0\). This isn't solved for \(x\) to find roots, but instead, we want to determine where the quadratic expression does not equal zero.

To do this, we factor the quadratic expression. Factoring converts it into the product of two binomials: \((x + 1)(x + 2)\). For this expression to equal zero, either \((x + 1) = 0\) or \((x + 2) = 0\). Solving these small linear equations gives us \(x = -1\) and \(x = -2\). Thus, \(x eq -1\) and \(x eq -2\), meaning these values are not part of the domain. This allows us to ensure that the quadratic function remains defined for the desired domain.

Factoring Expressions

Factoring is a mathematical process used to break down expressions into simpler products that, when multiplied together, yield the original expression. In the context of quadratic equations, factoring is a method to rewrite a quadratic in the form of \((x + m)(x + n)\). This form reveals the roots directly. The roots are the values of \(x\) that make each factor equal to zero.

For the quadratic expression \(x^2 + 3x + 2\), we factor it to become \((x + 1)(x + 2)\). Factoring is crucial because it allows us to determine the points where the quadratic expression is equal to zero, which are the roots \(-1\) and \(-2\). By recognizing these roots, we can exclude them from the domain of the function to ensure the function remains well-defined.

• Identify terms to factor: Look for common factors or apply patterns.
• Check using distributive property: Expand back to ensure accuracy.
• Use the factored form to find roots or restrictions as needed in the context of the problem.

Factoring expressions simplifies complex problems by breaking them into manageable parts.