Problem 131 - Algebra and Trigonometry

Question

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$ 3^{x}=2 x+3 $$

Step-by-Step Solution

Answer

The solution will depend on the exact x-coordinate of the intersection point. Once this value is obtained, substitute it into the original equation to verify it as a solution.

Step 1: Graph the Functions

Using any graphing utility, graph the two functions \(3^{x}\) and \(2x + 3\) in the same viewing rectangle. Be sure that the entire graphs are displayed. Note the point where the two graphs intersect.

Step 2: Determine the Intersection Point

The intersection point is the ordered pair \((x,y)\) that satisfies the equations of both graphs. By inspection, identify the x-coordinate of the intersection point, which will be your solutionset.

Step 3: Verification

Once you have determined the x-coordinate, you should substitute this value into the original equation to verify that it is indeed a solution. In this case, the value should be substituted into \(3^{x} = 2x + 3\) to confirm compatibility.

Key Concepts

Intersection Point

When solving equations graphically, the intersection point plays a crucial role. It's where the graphs of the two given functions meet. This point reflects both an x and y-coordinate, providing a solution applicable to the equations of both functions involved. In our exercise, the intersection point is found by comparing the graph of the exponential function, \(3^{x}\), and the linear equation \(2x + 3\).

• The x-coordinate of this intersection signifies the solution to the equation \(3^{x} = 2x + 3\).
• To find this point, you visually inspect or use the graphing utility's features to pinpoint where both graphs cross.

Identifying the intersection on the graph reveals the value where both sides of the equation equalize, guaranteeing their satisfaction of the equation's conditions.

Graphing Utility

A graphing utility is an essential tool for visualizing mathematical functions and solving equations graphically. This can be a graphing calculator or software that allows you to plot equations and functions effectively. Using a graphing utility helps to:

• Display multiple functions on the same set of axes, which is necessary to find their intersection points.
• Allow for zooming and adjusting views to precisely locate intersection points where functions meet or cross.

In our scenario, you input the equation parts \(3^{x}\) and \(2x + 3\) into the utility. This can immediately show where these two functions meet, simplifying the process of finding the solution to the equation by identifying the intersection point.

Exponential Functions

Exponential functions, such as \(3^{x}\), are functions where a constant base is raised to a variable exponent. They are distinguished by their rapid growth or decay depending on the context. Key characteristics of exponential functions to remember include:

• They always involve a constant base raised to a power that varies.
• For positive bases greater than 1, like 3 in \(3^{x}\), the function will increase as \(x\) increases.
• An asymptote is present, meaning the graph will not touch the x-axis at negative infinity.

In graphing contexts, these properties are essential for predicting the behavior of the graph and precisely locating intersections with other function types.

Linear Equations

Linear equations, like \(2x + 3\), are far more predictable and straightforward than nonlinear ones. They graph as straight lines, which makes their behavior easy to understand:

• They have a constant slope, defined by the coefficient of \(x\). For \(2x + 3\), the slope is 2, resulting in an upward slant.
• The y-intercept is where the line crosses the y-axis. For \(2x + 3\), this happens at the point (0,3).

These characteristics allow us to easily graph linear equations and look for their intersections with other types of functions, such as exponential ones. Understanding how these lines behave makes it easier to anticipate intersections when paired with exponential growth or decay.