91Ó°ÊÓ

In mathematics, the domain of a function refers to the set of "input" or argument values for which the function is defined. For example, the domain of the function \(f(x) = \sin^{-1}\left(\frac{1}{|x^2 - 11|}\right)\) is determined by finding values of \(x\) that do not make the function undefined.

Firstly, consider the condition where the denominator cannot be zero. Solve \(x^2 - 11 = 0\), resulting in \(x = \sqrt{11}\) and \(x = -\sqrt{11}\). These points are not in the domain, as they make the denominator zero.

Additionally, consider the range of the inverse sine function, which is only real-valued for inputs between -1 and 1, inclusive. Therefore, the expression inside sine inverse has to maintain \(-1 \leq \frac{1}{|x^2 - 11|} \leq 1\). This ensures that \(|x^2 - 11|\) is at least 1. Thus, the domain incorporates values where the function does not become undefined and fits within the proper sine inverse input range.

Quadratic equations are polynomial equations of degree 2, often represented as \(ax^2 + bx + c = 0\). Solving them can involve various methods including factoring, completing the square, or using the quadratic formula \(-\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

In the exercise, identifying where \(x^2 - 11 = 0\) is crucial because it reveals points where the function \(f(x)\) becomes undefined. Solving this specific equation yields \(x = \pm\sqrt{11}\).

This solution informs the domain by pinpointing values we must exclude to ensure the expression \(\frac{1}{|x^2 - 11|}\) does not involve division by zero. Quadratic equations thus play a key role in dealing with problems involving domain restrictions and other critical function behaviors.

Inequalities compare values, expressions, or equations using signs like \(<\), \(>\), \(\leq\), or \(\geq\). They are vital in determining the set of values that satisfy certain conditions, especially when finding function domains.

In the context of our exercise, inequalities help establish ranges for \(x\) such that the expression remains valid within the inverse sine's permissible range. The condition \(-1 \leq \frac{1}{|x^2 - 11|} \leq 1\) ensures that input into \(\sin^{-1}\left(...\right)\) results in a real number.

Inequalities prompt us to solve \(|x^2 - 11| \geq 1\). It essentially splits into two parts: \(x^2 - 11 \geq 1\) and \(x^2 - 11 \leq -1\). Both these solutions give us a range of values for \(x\) that satisfy the conditions for the function to be defined and real-valued.