- Write the expression for the quotient: \(\frac{x^2 - x - 2}{x - 2}\).
- Factor the numerator: \(x^2 - x - 2 = (x - 2)(x + 1)\).
- Simplify the quotient by canceling the common factor: \((x + 1)\).
- Identify the domain restriction by setting the denominator to zero: \(x
eq 2\).
- Final expression: \(\left(\frac{f}{g}\right)(x) = \boxed{x + 1}\), with domain \(x
eq \boxed{2}\).
Explanation
- Defining the Quotient Function
To find the quotient function \(\left(\frac{f}{g}\right)(x)\), we first define the quotient as the ratio of the two given functions:
\[\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 - x - 2}{x - 2}\]
The functions \(f(x) = x^2 - x - 2\) and \(g(x) = x - 2\) are both polynomials, and thus their natural domains are all real numbers, \(\mathbb{R}\). However, the domain of the quotient function is restricted by the condition that the denominator cannot be zero.
- Factoring the Numerator
To simplify the fraction, we factor the quadratic expression in the numerator. We are looking for two numbers that multiply to \(-2\) and add to \(-1\). These numbers are \(-2\) and \(1\). Thus, the quadratic can be factored as follows:
\[x^2 - x - 2 = (x - 2)(x + 1)\]
Substituting this back into our expression for the quotient gives:
\[\left(\frac{f}{g}\right)(x) = \frac{(x - 2)(x + 1)}{x - 2}\]
- Simplifying and Determining the Domain
For all values of \(x\) where \(x - 2
eq 0\), we can cancel the common factor \((x - 2)\) from the numerator and the denominator:
\[\left(\frac{f}{g}\right)(x) = x + 1\]
This simplification is valid as long as \(x - 2
eq 0\), which means \(x
eq 2\). Therefore, the domain of the function is all real numbers except \(x = 2\).
Examples
Imagine you are calculating the average profit per item sold. If the total profit is modeled by the function \(f(x) = x^2 - x - 2\) and the number of items sold is modeled by \(g(x) = x - 2\), the average profit per item is the ratio \(\frac{f(x)}{g(x)}\). Simplifying this expression allows you to quickly evaluate the profit trends for different quantities, provided you don't divide by zero (which would happen if you sold exactly 2 items). This is a common application of rational function simplification in business and economics to analyze performance metrics.