Problem 42 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution is approximately \(x = 1.42\).

Step 1: Understand the Problem

The given equation is \( 1+\sqrt{x}=\sqrt{1+x^{2}} \) and we are asked to solve it graphically over the interval \([-1,5]\). Our task is to find the values of \(x\) that satisfy this equation.

Step 2: Rearrange the Equation

Rewrite the equation in a form suitable for graphical analysis. Define two separate functions: \( f(x) = 1 + \sqrt{x} \) and \( g(x) = \sqrt{1 + x^2} \). Our goal is to find where these two functions intersect over the interval \([-1, 5]\).

Step 3: Graph the Functions

Plot the functions \( f(x) \) and \( g(x) \) on the same graph. Use appropriate graphing software or a graphing calculator to accurately display both functions over the specified interval.

Step 4: Identify Intersection Points

Locate the points where the two graphs intersect. These intersections represent the solutions to the equation \( 1 + \sqrt{x} = \sqrt{1 + x^2} \) within the given interval.

Step 5: Find the x-coordinates of Intersections

Read off the \(x\)-coordinates of the intersection points from the graph. Ensure these are precise to two decimal places as required by the problem statement.

Key Concepts

Graphing Functions

Graphing functions is a fundamental method in mathematics that helps visualize complex relationships between variables. In the case of solving equations graphically, you are essentially breaking down the equation into simpler components. For our equation, we rewrite it as two separate functions:

• \(f(x) = 1 + \sqrt{x}\)
• \(g(x) = \sqrt{1 + x^2}\)

By visualizing these functions on a graph, we get a clear picture of how they behave and interact across various values of \(x\). This is particularly useful as it allows us to see where they overlap or intersect, which will help us solve the equation graphically. Remember, it's not just about pinpointing solutions, it's about understanding the overall trends of the functions.

Intersection Points

Intersection points are key in graphically solving equations. These are the points on the graph where two or more functions meet. For our problem, this means finding the \(x\)-coordinate(s) where \(f(x)\) and \(g(x)\) are equal. Why are they essential?

• They tell us where the two functions balance each other out, meaning their outputs are the same.
• This equals the solution of the equation \(1+\sqrt{x} = \sqrt{1 + x^2}\).

To determine these points, you need to carefully analyze the graph in the given interval \([-1,5]\). Make sure you double-check these graphically derived intersection points, since accuracy to two decimal places is crucial for the solution.

Function Analysis

Function analysis involves examining the behavior of mathematical functions over certain intervals. By analyzing both \(f(x)\) and \(g(x)\) in our problem, you ensure a thorough understanding of the equation's dynamics. Here's what function analysis usually entails:

• Domain and Range: Evaluate where each function is defined and its possible output values. For instance, \(f(x)\) starts at \(x=0\) since the square root of a negative number is not real, hence doesn't apply in the context.
• Behavior: Check how the functions increase or decrease. This helps in making predictions about where intersections might naturally occur or not occur at all.

Such an analysis gives insight into the optimal method and places for graphing, forming a foundational step before actual plotting.

Graphing Calculator

A graphing calculator is a vital tool for plotting functions and finding solutions graphically. When handling equations like \(1 + \sqrt{x} = \sqrt{1 + x^2}\), graphing calculators provide precise graphical representations that are hard to match manually. They offer several advantages:

• Accuracy: Ability to plot functions over specified intervals (like \([-1,5]\)) with precision.
• Convenience: Easier manipulation and adjustment of functions to observe changes dynamically.
• Interactivity: Some calculators allow you to trace and zoom to pinpoint exact intersection coordinates, ensuring results like "to two decimal places" are achievable.

Investing time in learning how to efficiently use a graphing calculator can save time and improve your mathematical intuition.

Mathematical Intervals

Understanding mathematical intervals is essential when graphically solving equations. An interval like \([-1,5]\) defines the specific segment of the \(x\)-axis to focus on. Here’s why intervals matter:

• They establish a clear boundary for your investigation, helping to avoid infinite or undefined values.
• Focusing ensures you don’t graph unnecessary parts of the functions, which simplifies analysis and correction.
• Intervals align with real-world constraints, like time limits or resource bounds, mirroring real-life mathematical applications.

By keeping the interval in mind, you ensure your graph includes all necessary data points for accurate solutions, while fully complying with the problem’s requirements.