Problem 101 - A Graphical Approach to College Algebra

Question

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=\ln (x+2)$$

Step-by-Step Solution

Answer

The solution to \( f(x)=0 \) is \( x=-1 \); \( f(x)<0 \) for \( x \in (-2, -1) \); \( f(x)\geq 0 \) for \( x \in [-1, \infty) \).

Step 1: Solve the Equation

To solve the equation \( f(x) = 0 \) where \( f(x) = \ln(x+2) \), we set \( \ln(x+2) = 0 \). \( \ln(a) = 0 \) implies \( a = 1 \), so \( x + 2 = 1 \). Solving for \( x \), we get \( x = -1 \).

Step 2: Graph the Function

Graph the function \( y = \ln(x+2) \). The function is defined for \( x > -2 \), has a vertical asymptote at \( x = -2 \), and passes through the point \( (-1, 0) \) because \( \ln(1) = 0 \). The graph increases continuously and approaches infinity as \( x \to \infty \).

Step 3: Analyze the Inequality \( f(x) < 0 \)

On the graph, observe where \( y = \ln(x+2) \) is below the x-axis. This occurs when \( x < -1 \) because \( \ln(x+2) \) is negative for values \( x \in (-2, -1) \).

Step 4: Analyze the Inequality \( f(x) \geq 0 \)

Inspect the graph where \( y = \ln(x+2) \) is on or above the x-axis. This happens for \( x \geq -1 \). Therefore, \( f(x) \geq 0 \) for \( x \in [-1, \infty) \).

Key Concepts

Natural Logarithm Functions

Natural logarithms are a fundamental concept in mathematics, and they're expressed using the notation \( \ln(x) \). This is the logarithm to the base \( e \), where \( e \approx 2.71828 \), which is a mathematical constant known as Euler's number. Natural logarithms are used extensively in calculus and science because they have unique properties and simple derivatives.
Understanding the function \( f(x) = \ln(x+2) \) involves knowing that:

• \( \ln(a) = 0 \) means that \( a = 1 \), since \( e^0 = 1 \).
• The domain of \( \ln(x+2) \) is for \( x > -2 \), as you cannot take the logarithm of a non-positive number.
• The function is continuous and always defined for positive inputs, where it increases without bound as the input increases.

Knowing these principles helps solve equations involving natural logarithms accurately and gives insight into how functions behave graphically.

Graphing Inequalities

Graphing inequalities is a way to visually represent solutions to equations or conditions involving inequality symbols, like \(<\) or \(\geq\). When you deal with the function \( y = \ln(x+2) \), graphing is an invaluable tool for understanding where the function's values meet specific inequality conditions.
Here's how you approach it for the function \( f(x) = \ln(x+2) \):

• Plot the function and note significant features like intersections with the x-axis. For this function, it crosses the x-axis at \( x = -1 \).
• Identify the intervals where the function is negative by checking where the graph sits below the x-axis. In our case, this is for \( x < -1 \).
• Find out where the function is non-negative by observing the graph's position on or above the x-axis. This happens when \( x \geq -1 \).

The graph provides a clear illustration of solution intervals and allows for a deeper understanding of each inequality's domain. Graphing turns abstract algebraic solutions into concrete visual data.

Analytical Methods

Analytical methods involve using algebraic manipulation and mathematical reasoning to solve equations and inequalities systematically. When solving \( f(x) = 0 \) for \( f(x) = \ln(x+2) \), each term is manipulated according to the rules of logarithms and algebra.

• Start by setting the logarithmic expression equal to zero as a way to balance the equation, \( \ln(x+2) = 0 \).
• Apply the property \( \ln(a) = 0 \) implies \( a = 1 \), determining \( x + 2 = 1 \).
• Solve the equation algebraically by isolating \( x \), which gives us \( x = -1 \).

Using analytical methods ensures a precise solution, providing necessary insight into the structure and behavior of the function. They provide the groundwork upon which a deeper understanding of the equations and the graph is based, ensuring a student can transition from algebra to visual interpretations seamlessly.