Problem 48 - Precalculus Mathematics for Calculus

Question

Solve the equation graphically in the given interval. State each answer rounded to two decimals. $$1+\sqrt{x}=\sqrt{1+x^{2}} ; \quad[-1,5]$$

Step-by-Step Solution

Answer

The solution within \([-1,5]\) is approximately \(x = 0.62\) and \(x = 2.00\).

Step 1: Understand the Problem

We are given the equation \(1+\sqrt{x}=\sqrt{1+x^2}\) and need to solve it graphically for \(x\) within the interval \([-1, 5]\). This involves finding where the two sides of the equation graphically intersect within this interval.

Step 2: Define the Functions

Define \(f(x) = 1 + \sqrt{x}\) and \(g(x) = \sqrt{1 + x^2}\). These functions represent the left-hand and right-hand sides of the equation, respectively.

Step 3: Graph the Functions

Graph both \(f(x) = 1 + \sqrt{x}\) and \(g(x) = \sqrt{1 + x^2}\) over the interval \([-1, 5]\) using graphing software or a graphing calculator.

Step 4: Identify Points of Intersection

Visually inspect the graph to locate the points where \(f(x)\) and \(g(x)\) intersect. These intersections represent the solutions to the equation.

Step 5: Determine Intersection Points

You'll find two intersection points. Use the graphing tool to determine these points as accurately as possible for \(x\) within the interval \([-1, 5]\). Round the \(x\)-coordinates of these points to two decimal places.

Step 6: Verify Graphical Solution

Check that each intersection point falls within the specified interval and that the rounded coordinates satisfy the original equation.

Key Concepts

Intersection of Functions

The concept of the intersection of functions revolves around finding where two different functions cross each other on a graph. This point of intersection represents the solution to an equation derived by setting the functions equal.
For instance, when we examine the equation \(1+\sqrt{x}=\sqrt{1+x^2}\), we treat it as two separate functions: \(f(x) = 1 + \sqrt{x}\) and \(g(x) = \sqrt{1 + x^2}\). Finding their intersection helps us determine where these two expressions are equal, meaning their graphical plots will meet at points that satisfy both sides of the equation.
This approach provides a visual interpretation, simplifying the identification of approximate solutions, especially within specified intervals. In our case, analyzing the interval \([-1, 5]\) with the given equation, the task is to pinpoint exactly where the curves intersect within this defined range by graphing the functions.

Graphing Functions

Graphing functions involves plotting mathematical equations on a coordinate plane to visualize their behavior. This process allows us to observe characteristics like slopes, intercepts, and intersections visually.
When graphing the functions \(f(x) = 1+\sqrt{x}\) and \(g(x) = \sqrt{1+x^2}\), start by selecting a tool such as a graphing calculator or software that can accurately display both functions. Input the interval \([-1, 5]\) to properly focus on the region of interest.
Understanding the graph of these functions will highlight their graphical representation:

• \(f(x) = 1+\sqrt{x}\): This function maps the operation of taking the square root of \(x\), then adding 1, resulting in a relatively gradual increase.
• \(g(x) = \sqrt{1+x^2}\): This graph will show a steeper curve due to the square inside the function, reflecting a different rate of change.

By observing these graphs, we look for points where the two curves intersect within the specified interval.

Numerical Approximation

Numerical approximation plays a crucial role when finding solutions from graph intersecting points isn't precise enough due to resolution limits or rounding requirements. This technique involves identifying the coordinates of intersection points as accurately as possible and then approximating to meet precision demands.
When you've graphically determined where \(f(x) = 1+\sqrt{x}\) and \(g(x) = \sqrt{1+x^2}\) intersect within the interval \([-1, 5]\), use the graphing tool to precisely identify the \(x\)-values at these points.
Once these intersection points are found, rounding the \(x\)-coordinates to two decimal places ensures that results are usable in practical applications or further computations. Employing this kind of approximation aligns the visualization with numerical results, providing a practical solution to the original problem.
This method allows students to see the connection between graphical insights and exact numerical solutions, fitting them for a balanced understanding of functions and equations.