Problem 61 - Algebra and Trigonometry

Question

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ 4^{-x}=\sqrt{x} $$

Step-by-Step Solution

Answer

The solutions are approximately \(x \approx -0.77\) and \(x \approx 0.25\).

Step 1: Understand the Equation

We need to find the x-values that satisfy the equation \[4^{-x} = \sqrt{x}\]This means we are looking for points where the function \(f(x) = 4^{-x}\) intersects the function \(g(x) = \sqrt{x}\).

Step 2: Set Up the Graphing Device

Use a calculator or graphing software to input the two functions. Enter \(y_1 = 4^{-x}\) and \(y_2 = \sqrt{x}\) separately so that they can be plotted on the same graph for visual comparison.

Step 3: Plot the Functions

After inputting the functions, plot them over a reasonable domain. Consider trying an x-range from -5 to 5 to ensure that the relevant points of intersection are visible on the graph.

Step 4: Identify the Intersection Points

Look for points where \(y_1\) and \(y_2\) intersect. Using the graph, you should be able to visually pinpoint where the graphs cross, or you can use a function on the graphing tool to find precise intersection points.

Step 5: Calculate the Intersection Points

To find intersection points accurately, use the graphing device’s feature to calculate them. This often involves selecting both graphs and providing a range where the intersection might occur. The tool should output the intersection points.

Step 6: Extract Numerical Solutions

The graphing device should provide solutions correct to two decimal places. For this equation, the approximate solutions are: \[x \approx -0.77\] and \[x \approx 0.25\].

Key Concepts

Intersection Points

To solve an equation like \[4^{-x} = \sqrt{x}\]you're essentially looking for the intersection points of two functions on a graph. Intersection points are where both functions have the same value for a given \(x\). This means they 'cross' or meet at that particular \(x\) value on the graph.

• In this exercise, we are finding the intersection of \(f(x) = 4^{-x}\) and \(g(x) = \sqrt{x}\).
• By graphing these functions, the points of intersection are the solutions to the equation.

Knowing this gives you a visual representation of where the solutions might be. Once you understand the concept of intersections, it becomes much easier to know why these specific \(x\)-values matter—they make both sides of the equation equal.

Graphing Technology

Graphing technology helps visualize mathematical problems like equations that may not be easily solved algebraically. These tools plot functions to show their shapes and relationships, like intersections.

• Graphing calculators or software allow you to input functions such as \(4^{-x}\) and \(\sqrt{x}\).
• The technology then plots both functions so you can see where they intersect.
• This visual aid is invaluable for solving complex equations, especially when there's more than one solution.

To use graphing technology effectively, ensure you enter each function correctly and choose a suitable range for \(x\) values so you don't miss any intersection points. This can be set by adjusting the view window, often spanning an \(x\)-range where you anticipate the functions meeting. With graphing technology, finding intersection points becomes a more straightforward task.

Decimal Approximation

In math, sometimes solutions can't be expressed as neat whole numbers or simple fractions. In these cases, decimal approximations are used. For a given mathematical problem, the solutions can express how precise you want to be.

• The exercise specifies finding the \(x\)-values to two decimal places, ensuring precision without unnecessary detail.
• This is handy when using tools like graphing calculators, which calculate these approximations automatically.
• Approximation is about balance: accuracy sufficient for purpose without cluttering results with excessive digits.

For example, the solutions \(x \approx -0.77\) and \(x \approx 0.25\) indicate we're keeping the final answers concise and clear. These values tell us precisely at which point the two graphs intersect, as accurately as our scenario requires.