Problem 46 - College Algebra

Question

For the following exercises, graph the inequality. $$y \geq e^{x}$$ $$\quad\quad y \leq \ln (x)+5$$

Step-by-Step Solution

Answer

Shade the region above \( y = e^x \) and below \( y = \ln(x) + 5 \).

Step 1: Graph the function y = e^x

First, we graph the function \( y = e^x \). This is an exponential function that grows rapidly from left to right. Plot a few key points such as \((0, 1)\), \((1, e)\), and \((-1, 1/e)\). Draw the curve through these points.

Step 2: Graph the function y = ln(x) + 5

Now graph the function \( y = \ln(x) + 5 \). This is a logarithmic function shifted 5 units up. Key points like \((1, 5)\) and \((e, 6)\) can help plot this curve. It approaches negative infinity as \(x\) approaches zero and increases slowly as \(x\) increases.

Step 3: Shading for the inequality y ≥ e^x

For the inequality \( y \geq e^x \), shade the region above or on the curve \( y = e^x \). This includes all points where the y-values are greater than or equal to the function values of \( e^x \).

Step 4: Shading for the inequality y ≤ ln(x) + 5

For the inequality \( y \leq \ln(x) + 5 \), shade the area below or on the curve \( y = \ln(x) + 5 \). This includes all points where the y-values are less than or equal to the function values of \( \ln(x) + 5 \).

Step 5: Identify the Intersection Region

The solution to the system of inequalities is the overlapping shaded region from the two previous steps. Check and highlight the area on the graph where the two shaded regions intersect. This region contains all the points that satisfy both inequalities \( y \geq e^x \) and \( y \leq \ln(x) + 5 \).

Key Concepts

Exponential Functions

Exponential functions are a crucial concept in mathematics, characterized by their rapid growth or decay. In the function \( y = e^x \), the base is the irrational number \( e \), approximately equal to 2.718. This function exhibits sharp growth, as demonstrated by plotting points such as \((0, 1)\), \((1, e)\), and \((-1, 1/e)\). Exponential functions are essential in various real-world applications, including compound interest calculations, population growth models, and decay processes like radioactive decay. Understanding the behavior of exponential functions is key to analyzing how systems evolve over time.

• Exhibits rapid growth or decay
• Base \( e \) is approximately 2.718
• Important in finance and natural sciences

Recognizing these characteristics allows students to graph these functions effectively, paving the way to explore and solve complex mathematical problems involving inequalities.

Logarithmic Functions

Logarithmic functions are inverses of exponential functions. The function \( y = \ln(x) + 5 \) is the result of shifting the basic logarithmic function \( y = \ln(x) \) 5 units upward. This shift moves the graph vertically, changing the y-intercepts but maintaining the shape. Logarithmic functions grow more slowly compared to exponential functions. They are important for understanding processes where growth diminishes over time, such as sound intensity or the Richter scale for earthquakes.

• Inverse of exponential functions
• Represent growth that slows over time
• Practical in analyzing scientific and real-world phenomena

When graphing \( y = \ln(x) + 5 \), it's vital to note that the x-values must be positive. Logarithmic functions decline indefinitely as they approach zero on the x-axis, representing their inverse nature compared to exponential growth.

Inequality Solutions

Solving inequalities involving exponential and logarithmic functions can be challenging, requiring an understanding of the graphs and regions defined by each inequality. For the inequality \( y \geq e^x \), the solution includes all points on or above the plotted curve \( y = e^x \). Conversely, \( y \leq \ln(x) + 5 \) encompasses all points on or below the curve \( y = \ln(x) + 5 \). Solutions to these inequalities lie at the intersection of the shaded regions on a graph.

• Solutions define regions on graphs
• Find overlapping areas for combined inequalities
• Represents all points satisfying both conditions

When graphing these solutions, the overlap of shaded areas represents the comprehensive solution where both inequality conditions are met. Careful plotting and shading of these areas are required for accuracy and proper representation.

Graphing Techniques

Graphing is a powerful way to visualize solutions to inequalities and understand the complex relationships between different types of functions. With exponential and logarithmic functions, correctly plotting the curve and understanding key features like asymptotes and intercepts are fundamental. Techniques involve plotting key points, such as intercepts, and carefully drawing curves through these points. For exponential functions, noticing the rapid increase is crucial, while for logarithmic functions, focus on the gentle slope and approach of the y-axis as \( x \) grows.

• Use key points to plot accurate graphs
• Understand curves and their interactions
• Shading identifies solution regions

Shading techniques help identify solution areas clearly by indicating whether the region falls above or below the given curve. This visual approach aids in determining the feasible region, reinforcing an understanding of inequality solutions through clear, graphical representation.