Problem 60 - Algebra and Trigonometry

Question

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ 2^{-x}=x-1 $$

Step-by-Step Solution

Answer

The solution is approximately \( x = 0.61 \).

Step 1: Rewrite the Equation for Graphing

The equation given is \( 2^{-x} = x - 1 \). We need to graph two functions: \( y = 2^{-x} \) and \( y = x - 1 \). These two graphs will help us visualize where the solution(s) to the equation are.

Step 2: Set Up the Graph

Using a graphing calculator or software, plot the two functions: \( y = 2^{-x} \) and \( y = x - 1 \). Choose a domain (e.g., from \( x = -5 \) to \( x = 5 \)) to ensure we can see the intersection point(s) on the graph.

Step 3: Identify the Intersection Points

Observe where the graph of \( y = 2^{-x} \) intersects the graph of \( y = x - 1 \). The intersection points are the solutions to the equation \( 2^{-x} = x - 1 \).

Step 4: Read the Intersection Coordinates

From the graph, find the \( x \)-coordinate of the intersection point. Make sure to note this value accurately, to two decimal places, to get the exact solution.

Step 5: Verify the Solution

Substitute the identified \( x \) value back into the original equation \( 2^{-x} = x - 1 \) to check if both sides are approximately equal, confirming the intersection point as a solution.

Key Concepts

Graphing Calculator

Using a graphing calculator can simplify the task of solving equations through visualization. These devices allow you to plot the functions on a graphical interface, making it easier to understand mathematical concepts. When dealing with the equation \( 2^{-x} = x - 1 \), the graphing calculator helps you to set up two function plots: \( y = 2^{-x} \) and \( y = x - 1 \). By observing the intersection of these graphs, you can determine the solution to the equation. To effectively use a graphing calculator, follow these steps:

• Input each function separately into the calculator.
• Adjust the viewing window by setting the domain, such as from \( x = -5 \) to \( x = 5 \), to see where the graphs intersect.
• Use the trace or intersection feature to precisely find the solution point where both graphs meet.

This process provides a visual method to solve equations that can be especially helpful for students who struggle with abstract algebraic solutions.

Intersection Points

Finding intersection points is crucial in solving equations graphically. In the context of our equation \( 2^{-x} = x - 1 \), intersection points represent values of \( x \) where both functions \( y = 2^{-x} \) and \( y = x - 1 \) are equal.

• To find these points, first, graph both equations on the same set of axes.
• Look for the point(s) where the two graphs intersect each other.

Once the intersection is identified, the \( x \)-coordinate of this intersection is the solution to the equation. This means that substituting this \( x \) value into both expressions on either side of the equation should yield the same result. Always cross-check your solution by substituting back into the original equation to ensure accuracy.

Exponential Functions

Exponential functions, like \( y = 2^{-x} \), are critical in the world of mathematics. They have unique characteristics that make them useful in various fields, including computer science and finance.
The function \( 2^{-x} \) decreases rapidly as \( x \) increases, demonstrating the exponential decay behavior.

• Exponential functions are typically of the form \( y = a^x \), where \( a \) is a positive constant greater than 1.
• When the exponent is negative, as in \( 2^{-x} \), the function depicts a decay.
• These functions approach zero but never get to zero, allowing for interesting behaviors on a graph.

Understanding these properties can assist in recognizing how these functions will look on a graph, and how they will interact with linear functions, like in our problem.

Linear Functions

Linear functions, such as \( y = x - 1 \), are simpler and graph as straight lines. These functions are foundational in algebra and appear frequently in various applications.
The key characteristics of linear functions include:

• The graph of a linear function is always a straight line.
• The general form of a linear function is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• For \( y = x - 1 \), the slope \( m \) is 1, indicating a 45-degree angle to the axes, and the y-intercept \( b \) is -1.

When graphed, this line will intersect with other types of functions, such as the exponential function in our case, at points that are solutions to the equation. Recognizing these interactions is crucial in graphing and provides a visual strategy for solving equations.