Problem 101 - A Graphical Approach to Precalculus with Limits

Question

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 \( x=-1 \). For \( f(x)<0 \), \( -2

Step 1: Solve the Equation

To solve the equation \( f(x) = 0 \), where \( f(x) = \ln(x+2) \), set \( \ln(x+2) = 0 \). Recall that \( \ln(a) = 0 \) implies \( a = 1 \). Thus, \( x + 2 = 1 \). Solve for \( x \) to get \( x = -1 \).

Step 2: Identify the Domain and Analyze the Function

Since \( f(x) = \ln(x+2) \), the domain is where \( x+2 > 0 \), or equivalently, \( x > -2 \). The function is defined for \( x > -2 \). Additionally, \( f(x) \) is increasing because the natural logarithm is an increasing function.

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

Draw the graph of \( y = \ln(x+2) \). The function passes through the point \( x = -1 \), where it equals zero. For \( x > -2 \) and \( x < -1 \), \( \ln(x+2) \) is negative. Thus, the solution to \( f(x) < 0 \) is \( -2 < x < -1 \).

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

Using the graph again, \( f(x) \) is zero at \( x = -1 \) and positive for \( x > -1 \). The solution to the inequality \( f(x) \geq 0 \) is \( x \geq -1 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are expressions where the logarithm, usually denoted as \( \ln \) or \( \log \), is the main operator. In this context, we deal with the natural logarithm function \( f(x) = \ln(x+2) \). The natural logarithm function is continuous and defined for any value where its argument is greater than zero.
When solving \( f(x) = 0 \), you're finding the point where the logarithm equals zero. Because \( \ln(a) = 0 \) when \( a = 1 \), we set \( x+2 = 1 \), leading to \( x = -1 \). This value is crucial, as it is where the function crosses the x-axis on its graph.

• The function increases as x increases, due to the properties of the logarithm.
• The principal attribute is the domain: \( x + 2 > 0 \) implies \( x > -2 \).

Inequalities

Understanding inequalities involves identifying regions on the graph where the function is either greater than or equal to zero, or less than zero. With logarithmic functions, you must always pay special attention to the domain.
To solve \( f(x) < 0 \), we need the points where the graph lies below the x-axis. This for \( \ln(x+2) < 0 \) occurs between \(-2 < x < -1\). Here, \( \ln(x+2) \) is negative because the argument is greater than 0 but less than 1.
For \( f(x) \geq 0 \), we're interested in where the graph is on or above the x-axis. This happens at \( x \geq -1 \), since \( \ln(x+2) \) is zero at \( x = -1 \) and positive for all x greater than \(-1\).

• Always consider the function's domain when solving inequalities.
• Check whether the inequality includes equal value at boundaries or only greater/less values.

Graph Analysis

Graph analysis of a function provides visual insight into its behavior, significantly enhancing understanding of equations and inequalities. In the case of \( y = \ln(x+2) \), the graph helps to visualize the point where \( f(x) = 0 \) and regions where \( f(x) < 0 \) or \( f(x) \geq 0 \).
The graph passes through the x-axis at \( x = -1 \), indicating the solution to \( f(x) = 0 \). To each side of this point, the graph varies:

• To the left, from \( -2 < x < -1 \), the graph dips below the x-axis, indicating \( f(x) < 0 \).
• To the right, from \( x = -1 \) onwards, the graph stays above the x-axis, confirming \( f(x) \geq 0 \).

The graphical representation also emphasizes how \( f(x) \) is undefined for \( x \leq -2 \), reinforcing the domain limitations inherent to logarithmic functions. Graphs transform abstract algebraic solutions into more tangible, visual solutions, enriching comprehension.

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Problem 102

Other exercises in this chapter

Problem 100

Solve \(-3^{x}+7=0\) for \(x,\) expressing \(x\) in terms of a base 3 logarithm.

View solution

Problem 100

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)

View solution

Problem 102

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)

View solution

Problem 103

The following equations are identities because they are true for all real mumbers. Use properties of logarithms to simplify the expression on the left side of t

View solution

Key Concepts

Logarithmic Functions

Logarithmic functions are expressions where the logarithm, usually denoted as \( \ln \) or \( \log \), is the main operator. In this context, we deal with the natural logarithm function \( f(x) = \ln(x+2) \). The natural logarithm function is continuous and defined for any value where its argument is greater than zero.
When solving \( f(x) = 0 \), you're finding the point where the logarithm equals zero. Because \( \ln(a) = 0 \) when \( a = 1 \), we set \( x+2 = 1 \), leading to \( x = -1 \). This value is crucial, as it is where the function crosses the x-axis on its graph.

• The function increases as x increases, due to the properties of the logarithm.
• The principal attribute is the domain: \( x + 2 > 0 \) implies \( x > -2 \).

Inequalities

Understanding inequalities involves identifying regions on the graph where the function is either greater than or equal to zero, or less than zero. With logarithmic functions, you must always pay special attention to the domain.
To solve \( f(x) < 0 \), we need the points where the graph lies below the x-axis. This for \( \ln(x+2) < 0 \) occurs between \(-2 < x < -1\). Here, \( \ln(x+2) \) is negative because the argument is greater than 0 but less than 1.
For \( f(x) \geq 0 \), we're interested in where the graph is on or above the x-axis. This happens at \( x \geq -1 \), since \( \ln(x+2) \) is zero at \( x = -1 \) and positive for all x greater than \(-1\).

• Always consider the function's domain when solving inequalities.
• Check whether the inequality includes equal value at boundaries or only greater/less values.

Graph Analysis

Graph analysis of a function provides visual insight into its behavior, significantly enhancing understanding of equations and inequalities. In the case of \( y = \ln(x+2) \), the graph helps to visualize the point where \( f(x) = 0 \) and regions where \( f(x) < 0 \) or \( f(x) \geq 0 \).
The graph passes through the x-axis at \( x = -1 \), indicating the solution to \( f(x) = 0 \). To each side of this point, the graph varies:

• To the left, from \( -2 < x < -1 \), the graph dips below the x-axis, indicating \( f(x) < 0 \).
• To the right, from \( x = -1 \) onwards, the graph stays above the x-axis, confirming \( f(x) \geq 0 \).

The graphical representation also emphasizes how \( f(x) \) is undefined for \( x \leq -2 \), reinforcing the domain limitations inherent to logarithmic functions. Graphs transform abstract algebraic solutions into more tangible, visual solutions, enriching comprehension.