Problem 48 - Algebra and Trigonometry

Question

\(43-50\) . Solve the equation graphically in the given interval. State each answer rounded to two decimals. $$ 1+\sqrt{x}=\sqrt{1+x^{2}} ;[-1,5] $$

Step-by-Step Solution

Answer

The graphs intersect around x = 0.68, indicating the solution.

Step 1: Identify Functions

Identify the functions represented in the equation: The left side of the equation is represented by the function \( f(x) = 1 + \sqrt{x} \). The right side of the equation is represented by the function \( g(x) = \sqrt{1 + x^2} \).

Step 2: Graph the Functions

Plot the functions \( f(x) \) and \( g(x) \) on a graph for the interval \([-1, 5]\). Use graphing software or a graphing calculator to visualize these two functions.

Step 3: Find the Intersection Points

Observe where the two graphs intersect within the interval \([-1, 5]\). The intersection points represent the values of \( x \) where the equation \( 1 + \sqrt{x} = \sqrt{1 + x^2} \) holds true.

Step 4: Approximate and Round the Intersection Points

From the graph, approximate the \( x \)-coordinates where the two graphs intersect. Ensure to round these values to two decimal places to obtain the solutions within the specified precision.

Key Concepts

Graphing Functions

Graphing functions is a visual way to understand mathematical relationships. Each function is like a special equation that describes a unique set of points on a graph. In our problem, we have two functions to consider:

• The first function, \( f(x) = 1 + \sqrt{x} \), rises slowly as \( x \) increases because it is a square root function combined with a constant.
• The second function, \( g(x) = \sqrt{1 + x^2} \), grows quicker as \( x \) increases due to the square root of a squared term plus one.

To graph these functions, you can use a graphing calculator or online graphing software. When you input each function, you'll get a curve that represents the function over the chosen interval, which in this case is from \(-1\) to \(5\). This visual representation is crucial for finding solutions graphically.

Intersection Points

Intersection points are where two graph lines (functions) cross each other. These points are important because they represent the solutions to an equation where the two functions are equal. In our example:

• To find the intersection, look at the graph and see where the curve of \( f(x) = 1 + \sqrt{x} \) meets the curve of \( g(x) = \sqrt{1 + x^2} \).
• It's within these intersection points that the values of \( x \) make both sides of our original equation true, thus solving the equation graphically.

Remember, these crossings are places where the ordered pairs \((x, y)\) have the same \( y \)-value for both functions at the same \( x \)-value.

Approximation

Approximation is necessary when calculating exact values directly from the graph is difficult. Graphs provide a visual representation and help us see where functions intersect, but they don't always give precise values for those \( x \)-coordinates. This is where approximation comes in. With a graph:

• You estimate the \( x \)-values where the functions intersect by observing closely.
• In many cases, like our problem, you might need to adjust your estimates to be as close as possible to the true intersection points.

Approximations are often required as graphs are limited by resolution and can be difficult to read accurately enough without additional tools or calculations.

Rounding Numbers

Rounding numbers involves adjusting a number to a specified degree of precision, often to make calculations simpler. When you round, you're making the number easier to use at the cost of some precision.

• In our exercise, solutions are to be rounded to two decimal places.
• This is often the standard in scientific work where approximated values need to be concise yet accurate enough for practical use.

To round to two decimal places, look at the third decimal place:

• If it's 5 or more, increase the second decimal by one.
• If it's less than 5, leave the second decimal place as is.

Rounding ensures that solutions from graphical methods are presentable and usable, especially when communicating results.