Problem 44 - Algebra and Trigonometry

Question

Solve the equation graphically in the given interval. State each answer correct to two decimals. $$ x^{1 / 2}+x^{1 / 3}-x=0 ; \quad[-1,5] $$

Step-by-Step Solution

Answer

The solutions of the equation are approximately found by identifying the x-intercepts of the graph of \( f(x) \) on [0,5].

Step 1: Understand the Problem

We need to solve the equation \( x^{1/2} + x^{1/3} - x = 0 \) graphically within the interval \([-1, 5]\). To do this, we'll plot the function \( f(x) = x^{1/2} + x^{1/3} - x \) and identify where it crosses the x-axis. These x-axis crossings will represent the solutions to the equation.

Step 2: Set Up the Graph

Define the function \( f(x) = x^{1/2} + x^{1/3} - x \). Note that \( x^{1/2} \) and \( x^{1/3} \) are defined only for \( x \geq 0 \). Therefore, we will only plot the function over the interval \([0, 5]\).

Step 3: Plot the Function

Using graphing software or a graphing calculator, plot the function \( f(x) = x^{1/2} + x^{1/3} - x \) over the interval \([0, 5]\). Look for any x-intercepts, where the graph crosses the x-axis. Each x-intercept corresponds to a solution of the equation.

Step 4: Identify the Solutions

Observe the points where the plotted function crosses the x-axis. Record these x-values as they represent the solutions to \( f(x) = 0 \).

Step 5: Verify and Round the Solutions

Verify each solution by substituting back into the original function to check that it results in zero. Round each solution to two decimal places as required.

Key Concepts

Plotting Functions

Graphical equation solving begins by translating the functional equation into a visual format, namely a graph. To start, define your function as given, which in this case is \( f(x) = x^{1/2} + x^{1/3} - x \). This expression includes components like square roots and cube roots.When plotting, use a computational tool such as graphing software or a calculator to aid accuracy, especially over specific intervals. Here, we're focusing on the domain \([0, 5]\) since those terms \( x^{1/2} \) and \( x^{1/3} \) require non-negative values.

• Input the function into your graphing tool.
• Select the interval \([0, 5]\) for plotting, ensuring clarity.

The visual representation helps pinpoint solutions where the function hits zero, known as x-intercepts. An accurate plot is crucial to elucidating the problem-solving path.

X-Axis Intercepts

In the realm of graphical solutions, identifying the x-axis intercepts is vital, as these indicate the function's roots or solutions to the equation.Every point where the function \( f(x) \) meets the x-axis represents a value of \( x \) that solves our equation \( f(x) = 0 \). During plotting within the interval \([0, 5]\), observe closely for any zero-crossing.

• These intercepts are the solutions to our function. For this exercise, locate visually on your graph.
• They are found by checking when the plotted line intersects the x-axis.

The intercepts provide concrete solutions, which should subsequently be verified and rounded to the required precision level, typically two decimal places here.

Solution Verification

Once you have identified potential solutions as x-axis intercepts, the next task is solution verification. This ensures that the solutions are correct and precise.Substitute each x-intercept back into the original function: \( x^{1/2} + x^{1/3} - x \). If the result equals zero, the intercept is indeed a solution.

• Verify accuracy by reassessing the plotted graph and performing quick evaluations with the equation.
• Ensure no arithmetic errors are made during substitution.

Subsequent to verifying these x-values, round them to the nearest two decimal places as instructed. This helps meet problem requirements and aligns with standard mathematical precision.

Interval Analysis

In graphical equation solutions, analyzing the chosen interval is crucial, as it confines where solutions can exist.Our function \( f(x) = x^{1/2} + x^{1/3} - x \) was analyzed over the interval \([0, 5]\). It's important because

• The functional expressions involved have inherent domain restrictions, such as \( x^{1/2} \) and \( x^{1/3} \) that mandate \( x \geq 0 \).
• Limiting the interval helps avoid unnecessary calculations in areas where the function isn't defined or relevant.

Conducting this analysis ensures solutions are both valid and appropriately bound, focusing efforts on the most probable regions for accurate solutions.