Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \geq 3^{x}\\\&y \geq 2\end{aligned}$$

Step-by-Step Solution

Answer

Shade above \( y = 2 \) and above \( y = 3^x \), finding the overlapping region.

Step 1: Graph the Exponential Function

First, graph the exponential function \( y = 3^x \). Note that this function passes through the point (0, 1) because any number raised to the power of 0 is 1. It increases rapidly as \( x \) increases, curving upwards to the right. To represent \( y \geq 3^x \), shade the area above this curve.

Step 2: Graph the Horizontal Line

Next, graph the horizontal line \( y = 2 \). This line is parallel to the x-axis and passes through y = 2. For the inequality \( y \geq 2 \), shade the area above this line. Notice that the line itself is included as part of the solution set since \( y \) can equal 2.

Step 3: Identify the Solution Set

The solution set for the system of inequalities \( y \geq 3^x \) and \( y \geq 2 \) is the region of the graph where the shaded areas from both inequalities overlap. To find this region, observe where both above-the-line and above-the-curve areas coincide. This includes the region above the line \( y = 2 \) and above \( y = 3^x \).

Key Concepts

Understanding Exponential Functions

Exponential functions are intriguing mathematical models where a constant base is raised to a variable exponent. They demonstrate unique properties and behaviors that set them apart from linear or polynomial functions.
In our exercise, the function is represented as \( y = 3^x \). Key Characteristics of Exponential Functions:

• Starts at a positive value greater than zero when \( x = 0 \), hence it passes through the point \( (0,1) \) for the function \( 3^x \).
• As \( x \) increases, \( y \) grows rapidly, indicating exponential growth.
• The graph of an exponential function never touches the x-axis, as it asymptotically approaches zero when moving towards negative infinity on the x-axis.

For graphing purposes, understanding these properties helps in plotting the curve correctly.
In inequalities, such as \( y \geq 3^x \), shading above the curve indicates that any point in this region is part of the solution set, including the curve itself.

Defining the Solution Set

The solution set of a system of inequalities is the collection of points that satisfies all conditions posed by the inequalities. For our problem, we need to find where the solutions to both \( y \geq 3^x \) and \( y \geq 2 \) overlap.Insights into Solution Sets:

• The solution set is visually represented as a shaded region on the graph where conditions from all inequalities are met.
• In \( y \geq 2 \), all points on or above the horizontal line \( y = 2 \) are included.
• For \( y \geq 3^x \), the solution set consists of points on or above the exponential curve.

By finding the overlapping shaded areas, we determine the ultimate solution set. This ensures that every point in this region meets both inequality conditions simultaneously.

Utilizing Graphing Techniques

Graphing inequalities is a visual method that helps us interpret and solve these mathematical expressions effectively. Employing graphing techniques makes for an easier understanding of where solutions lie.Effective Graphing Techniques:

• Always start by drawing the lines or curves that represent each inequality's boundary conditions. Use dashed lines for strict inequalities (like \( > \) or \( < \)) and solid lines for inclusive ones (like \( \geq \) or \( \leq \)).
• In this scenario, because we have \( y \geq 3^x \) and \( y \geq 2 \), both the curve and the line are solid, meaning they are part of the solution set.
• Shade the regions that comply with each inequality. This makes it simple to identify regions that satisfy multiple inequalities by looking at overlapping areas.

Graphing these components together allows for a clear visualization of the solution set, ensuring every included point abides by the given inequalities.