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Chapter 4: Problem 104

Use a graphing calculator to find the solution set of each equation. Approximate the solution \((s)\) to the nearest tenth. $$x=2^{x}$$

Short Answer

Expert verified
The solution to the equation approximately is x 鈮� 0.6.

Step by step solution

01

- Enter the Equation

Input the equation into the graphing calculator. Enter the function as two separate expressions: y1 = x and y2 = 2^x.
02

- Graph the Equations

Graph both functions y1 and y2 on the same set of axes. Observe where the two graphs intersect, as these points represent the solutions to the equation.
03

- Locate the Intersection Points

Use the graphing calculator鈥檚 鈥業ntersect鈥� feature to find the intersection point(s) of the graphs of y1 and y2. This feature often requires you to select the two functions and provide an initial guess for the x-value of the intersection.
04

- Approximate the Solution

The calculator will provide the x-coordinate of the intersection with high precision. Round this x-coordinate to the nearest tenth to get the approximate solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graphing calculator
A graphing calculator is a powerful tool for solving equations, especially ones involving functions that aren't easy to manipulate algebraically. To solve the equation using a graphing calculator, follow these steps:
  • Start by turning on your graphing calculator and accessing the function input screen.
  • Enter the equations separately as y1 = x and y2 = 2x.
  • Ensure proper syntax; for example, some calculators require you to use a symbol like '^' for exponents.
After entering the equations, you're ready to move on to graphing them.
intersection points
Intersection points are where two graphs meet on a coordinate plane. These points are essential because they represent the solutions to the equation where y1 = y2.
  • To find the intersection points, you first need to graph the two equations.
  • On the graphing calculator, you'll often find this function under a menu labeled something like 'Calc' or 'Calculate.'
  • Select 'Intersect' and choose the lines y1 = x and y2 = 2x.
  • The calculator will ask you to provide an initial guess for the x-coordinate, which helps the calculator to narrow down the intersection point.
Carefully observing these steps ensures that you correctly identify where the graphs intersect.
approximate solutions
Approximate solutions are used when the exact value is hard to determine. After finding the intersection point on the graphing calculator, you'll likely receive a very precise value for x.
  • For instance, if the intersection point is found to be x = 0.696, you would round this to the nearest tenth.
  • In this case, the approximate solution is x = 0.7.
  • Remember to check the calculator's settings to ensure it's displaying enough decimal places for precision.
This approximation step is crucial in providing a usable solution that鈥檚 easier to interpret, especially in real-world applications.
exponential functions
Exponential functions, like y = 2x, involve a variable in the exponent. These functions grow much faster than linear functions like y = x.
  • The general form of an exponential function is y = ax where 'a' is a constant base and 'x' is the exponent.
  • In our specific problem, y2 = 2x, '2' is the base and 'x' is the variable exponent.
  • These functions are crucial in various fields, such as finance, biology, and physics, due to their rapid growth.
Understanding how exponential functions behave will help you grasp why intersections with other types of functions are often interesting and valuable.

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Most popular questions from this chapter

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