Find the domain of the following function. Do not use a graphing calculator.
$f(x)=\frac{1}{x^2-x-2}$
The domain is □ .
The domain of a rational function consists of all real numbers for which the expression is well-defined. For the function \(f(x) = \frac{1}{x^2 - x - 2}\), the expression is defined as long as the denominator is not equal to zero. Therefore, we must find the values of \(x\) that satisfy the equation \(x^2 - x - 2 = 0\) and exclude them from the set of all real numbers.
To solve the quadratic equation \(x^2 - x - 2 = 0\), we look for factors of \(-2\) that sum to \(-1\). The numbers \(-2\) and \(1\) satisfy these conditions, as \((-2) \times 1 = -2\) and \((-2) + 1 = -1\). Thus, we can factor the quadratic expression as follows:
\[x^2 - x - 2 = (x - 2)(x + 1).\]
Setting this equal to zero gives us:
\[(x - 2)(x + 1) = 0.\]
From the factored form \((x - 2)(x + 1) = 0\), we determine that the denominator is zero when \(x - 2 = 0\) or \(x + 1 = 0\). Solving these linear equations gives:
\[x = 2 \quad \text{and} \quad x = -1.\]
These are the two values for which the function \(f(x)\) is undefined.
Since the function is defined for all real numbers except \(x = -1\) and \(x = 2\), the domain consists of all real numbers except these two values. In interval notation, we represent this as the union of three intervals: the numbers less than \(-1\), the numbers between \(-1\) and \(2\), and the numbers greater than \(2\). Therefore, the domain is:
\[\boxed{(-\infty, -1) \cup (-1, 2) \cup (2, \infty)}.\]
The concept of a function's domain is crucial in fields like engineering and economics. For example, consider an electricity distribution function where the power supply might become infinite (a "singularity") at certain load points, causing system failure. Identifying the domain helps engineers determine the safe operating range for a network, ensuring that variables like voltage or load current stay within defined, manageable bounds. By avoiding these "undefined" points, they ensure the stability and safety of the electrical infrastructure.