Problem 109 - Algebra and Trigonometry

Question

In Exercises 109–112, find the domain of each logarithmic function. $$ f(x)=\ln \left(x^{2}-x-2\right) $$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\ln \left(x^{2}-x-2\right)\) is \((- \infty, -1) \cup (2, +\infty)\).

Step 1: Form the Quadratic Equation

First, set the function inside the natural log, \(x^{2} - x - 2 = 0\). This is a quadratic equation which can be solved easily.

Step 2: Solve the Quadratic Equation

Factor the equation to get \((x - 2)(x + 1) = 0\), which gives the critical points \(x_1 = 2\) and \(x_2 = -1\).

Step 3: Determine the Interval for the Solution

It's known that the sign of a quadratic function changes at its zeros. One should test each interval, \((- \infty, -1)\), \((-1, 2)\), and \((2, +\infty)\), by choosing a test point in each interval and then substituting it into \(x^{2} - x - 2\). If the test point satisfies \(x^{2} - x - 2 > 0\), then the entire interval satisfies the inequality.

Step 4: Test the Intervals

Choosing \(x = -2\) for the first interval, \(x = 0\) for the second interval, and \(x = 3\) for the third interval will yield \( (-2)^2 - (-2) - 2 = 4 > 0\), \( (0)^2 - 0 - 2 = -2 < 0\) and \( (3)^2 - 3 - 2 = 4 > 0\) respectively. Therefore, the intervals that satisfy the inequality are \((- \infty, -1)\) and \((2, +\infty)\).

Step 5: Write the Final Answer

Finally, combine the intervals to express the domain of the logarithmic function. It is \((- \infty, -1) \cup (2, +\infty)\).

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation. It is of the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). The quadratic equation used in our exercise is \( x^2 - x - 2 = 0 \). It defines the values of \( x \) that will make the quadratic expression zero.

Quadratic equations are important because they can be used to model various real-world situations like the path of an object in projectile motion. Solving a quadratic equation allows us to find the roots, or solutions, which are the x-values for which the expression equals zero. These roots also play a crucial role in defining domains and intervals, especially in the context of inequalities in logarithmic functions.

Factoring Quadratic Expressions

Factoring is one of the simplest methods for solving a quadratic equation. For our equation \( x^2 - x - 2 = 0 \), factoring helps break down the expression into simpler terms, making it easier to find the roots.

To factor a quadratic like \( x^2 - x - 2 \), you look for two numbers that multiply to the constant term (\(-2\)) and add up to the coefficient of the linear term (\(-1\)). These numbers are \(-2\) and \(+1\). Thus, the expression can be rewritten as \((x - 2)(x + 1) = 0 \).

Once factored, each factor can be set to zero:

• \(x - 2 = 0\) gives us \(x = 2\)
• \(x + 1 = 0\) gives us \(x = -1\)

These values are the roots of the quadratic equation. Knowing these roots allows us to test which intervals of \( x \) satisfy the inequality in the logarithmic function's domain.

Inequalities

Inequalities help us determine the range of values for a variable that will make a given expression true. For a quadratic expression like \( x^2 - x - 2 \), checking inequalities is crucial to finding the domain of the related logarithmic function, \( f(x)=\ln(x^2-x-2) \).

To find the domain, we need values of \( x \) that ensure \( x^2 - x - 2 > 0 \). We use test points in the regions defined by the roots \( x = -1 \) and \( x = 2 \). These intervals are:

• \((-\infty, -1)\)
• \((-1, 2)\)
• \((2, +\infty)\)

Testing points like \(x = -2\), \(x = 0\), and \(x = 3\) in these intervals helps us determine which intervals satisfy the inequality:
- \((-\infty, -1)\) because \(4 > 0\)
- \((2, +\infty)\) because \(4 > 0\)
Overall, the function \(\ln(x^2-x-2)\) is defined in the intervals \((-\infty, -1)\) and \((2, +\infty)\).

Natural Logarithms

Natural logarithms are logarithms with the base \(e\), where \(e\) is an irrational constant approximately equal to \(2.718\). When dealing with the function \(f(x) = \ln(x^2 - x - 2)\), understanding the properties of natural logarithms is essential because they are only defined for positive arguments.

This means, for \( \ln(x^2 - x - 2) \) to be valid, the expression inside the logarithm \( x^2 - x - 2 \) must be greater than zero. This is why finding the correct domain for \( x \) becomes critical. The roots \( x = -1 \) and \( x = 2 \) partition the real numbers into intervals, where some of them make the expression inside the log positive.

- If the result of the expression \( x^2 - x - 2 \) is positive, then \( \ln(x^2 - x - 2) \) yields a real number.

This ensures \((-\infty, -1)\) and \((2, +\infty)\) are the suitable intervals, constituting the domain where the function is defined and hence the way the natural logarithm is effectively used to understand the variation of the function.