In Exercises 109–112, find the domain of each logarithmic function. f(x) = ln (x² - x − 2)

Verified step by step guidance

1. Identify the expression inside the logarithm: $x^2-x-2$.
2. Set the expression inside the logarithm greater than zero: $x^2-x-2>0$.
3. Factor the quadratic expression: $(x-2)(x+1)>0$.
4. Determine the critical points by setting each factor equal to zero: $x-2=0$ and $x+1=0$, giving $x=2$ and $x=-1$.
5. Use a sign chart or test intervals around the critical points to find where the product $(x-2)(x+1)$ is positive, which will give the domain of the function.

Key Concepts

Logarithmic Functions

Logarithmic functions, such as f(x) = ln(x), are defined for positive arguments only. This means that the expression inside the logarithm must be greater than zero for the function to be valid. Understanding this property is crucial for determining the domain of any logarithmic function.

Finding the Domain

The domain of a function refers to the set of all possible input values (x-values) that will produce valid outputs. For logarithmic functions, this involves solving inequalities to find where the argument of the logarithm is positive. In this case, we need to solve the inequality x² - x - 2 > 0 to find the valid x-values.

Factoring Quadratic Expressions

Factoring is a method used to simplify quadratic expressions into products of binomials. For the expression x² - x - 2, factoring helps identify the roots of the equation, which are critical for determining intervals where the expression is positive or negative. This step is essential for accurately finding the domain of the logarithmic function.