Problem 49 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The solution is approximately at \( x = 0 \).

Step 1: Plot the Functions

Graph the functions \( y_1 = x^{1/3} \) and \( y_2 = x \) over the interval \([-3, 3]\). These are the functions on either side of the equation \( x^{1/3} - x = 0 \).

Step 2: Find Points of Intersection

The solution(s) to the equation are the x-values where both graphs intersect. Visually inspect the graph to identify these intersections along the x-axis within the interval.

Step 3: Determine Approximate Solutions

Round the x-values of the intersection points to two decimal places. Estimate these values from the graph's intersections for accuracy.

Key Concepts

Cubic Root Functions

Cubic root functions form an essential component of algebra and calculus. They involve finding the cube root, or \(x^{1/3}\), of a number. This means that we are searching for a number that, when multiplied by itself three times, gives the original number. The cubic root function is defined as:

• The principal root of \(x\), which is \(x^{1/3}\).
• Defined for all real numbers, including negative numbers.
• Provides a real result because cubing a number preserves its sign.

This property makes cubic root functions particularly interesting when graphing, as they appear symmetric with respect to the origin. They can pass smoothly from negative through zero to positive values, unlike square roots that are not defined for negative numbers within the real number system.

Intersection Points

Understanding intersection points is crucial for solving equations graphically. These points occur where two graphs meet or cross.
For the equation \(x^{1/3} - x = 0\), we consider two separate functions, \(y_1 = x^{1/3}\) and \(y_2 = x\).

• An intersection point is where \(y_1 = y_2\).
• Algebraically, this means the values of \(y\) at the point of intersection are the same for both functions.

Identifying where these points occur graphically allows us to estimate solutions to the equation. Each x-coordinate at these intersections represents a solution to the equation within the given interval. These intersections are vital as they visually and conceptually demonstrate the equality represented in the equation.

Graphing Techniques

Graphing techniques are essential for visually understanding and solving equations. To graph cubic root and linear functions:

• Start by selecting an appropriate interval, such as \([-3, 3]\) in this case.
• Plot key points on the graph by substituting certain x-values into the function to find corresponding y-values.
• Use these points to sketch the curve ensuring a smooth transition through these points for \(x^{1/3}\) and a straight line for \(x\).

Keep in mind that the cubic root function will gradually change and may not have drastic bends or curves within this range, providing more clarity on where the line and curve intersect in the visual plot. Use graphing techniques carefully to ensure precision when estimating the intersection points.

Function Plotting

Function plotting is the final piece of the puzzle in graphical equation solving. When plotting, it's important to:

• Ensure accuracy by checking each plotted point before connecting them.
• Use graphing software or a calculator for precise plots, particularly when handling more complex functions like cubic roots.
• Take note of the behavior of each function: the cubic root function \(y_1 = x^{1/3}\) forms a curve, while \(y_2 = x\) forms a straight line.
• Visually inspect the plot to determine the points of intersection.

By carefully plotting these functions, not only can you find the solution to the equation, but you can also gain a deeper understanding of how these functions behave and interact graphically.