Problem 5 - A Graphical Approach to Precalculus with Limits

Question

In Exercises begin by drawing a rough sketch to determine the number of real solutions for the equation \(y_{1}=y_{2}\) Then solve this equation by hand. Give the solution set and any extraneous values that may occur. Do not use a calculator. $$\begin{aligned} &y_{1}=\sqrt[3]{x}\\\ &y_{2}=x^{2} \end{aligned}$$

Step-by-Step Solution

Answer

The solution set is \( \{0, 1\} \). No extraneous values.

Step 1: Draw A Rough Sketch of the Functions

Let's sketch the graph of each function to visualize where they intersect. The graph of \( y_1 = \sqrt[3]{x} \) is a cube root function, which is symmetric about the origin and passes through points like \( (-1, -1), (0, 0), (1, 1) \). The graph of \( y_2 = x^2 \) is a parabola opening upwards with vertex at \( (0, 0) \). By sketching these, you can see they both intersect the origin and possibly have other intersection points as well.

Step 2: Set Equations Equal to Each Other

To find the points of intersection, set \( y_1 \) and \( y_2 \) equal to each other: \( \sqrt[3]{x} = x^2 \). Solve this equation to find \( x \)-values where both functions have the same \( y \)-value.

Step 3: Solve the Intersection Equation

Begin solving \( \sqrt[3]{x} = x^2 \). Raising both sides to the power of 3 to eliminate the cube root gives \( x = (x^2)^3 \). Simplify to obtain \( x = x^6 \).

Step 4: Simplify The Equation

Rearrange the equation from the previous step: \( x - x^6 = 0 \). Factor out \( x \) to get \( x(1 - x^5) = 0 \).

Step 5: Solve for x

Set each factor equal to zero:1. \( x = 0 \), an obvious solution since both graphs pass through the origin.2. Solve \( 1 - x^5 = 0 \) which gives \( x^5 = 1 \). Further simplify to get the real solution \( x = 1 \).

Step 6: Determine the Solution Set and Possible Extraneous Values

The solutions to \( \sqrt[3]{x} = x^2 \) are \( x = 0 \) and \( x = 1 \). Since both solutions make sense in the original functions, there are no extraneous solutions. Verify by substituting back into both functions to check they give the same \( y \)-values.

Key Concepts

Graphical Representation

Graphical representation is a powerful way to analyze equations visually. It involves plotting functions on a coordinate plane to understand their behavior and intersections. In the given exercise, we look at the equation \( y_1 = \sqrt[3]{x} \) and \( y_2 = x^2 \), plotting them allows us to see where they might intersect. By sketching, we notice that both graphs are centered around the origin and can visually identify points of intersection. This initial sketch helps us estimate the number of real solutions and offers a visual confirmation to solutions we derive algebraically. Sketching effectively simplifies complex equations by translating them into visual problems, aiding in comprehension.

Cube Root Function

The cube root function \( y = \sqrt[3]{x} \) represents numbers whose cube is equal to \( x \). Its graph is unique in that it resembles an elongated S-curve, symmetric about the origin, covering both positive and negative values of \( x \).

• The function passes through points like \((-1, -1)\), \((0, 0)\), and \((1, 1)\).
• It is always real and continuous for all real numbers.
• As \( x \) increases or decreases, the curve gradually approaches a straight line with a slope that indicates the rate of change.

This characteristic makes the cube root function particularly interesting, especially when compared to polynomial functions, providing distinct patterns of growth and decay.

Parabola

A parabola is a U-shaped curve found in quadratic equations, commonly appearing in the form \( y = ax^2 + bx + c \). In our exercise, we encounter the simple form \( y = x^2 \), which opens upwards with its vertex at the origin \((0, 0)\).

• Features of this parabola include symmetry around the y-axis and consistent growth as \( x \) moves away from zero in either direction.
• The graph's rate of increase is quadratic, meaning it doesn’t just rise linearly but accelerates as \( x \) values grow larger.
• It touches and crosses the cube root graph at well-calculated points of intersection.

Understanding the parabola's characteristics aids in determining how it interacts with other functions, facilitating intersections that reveal real solutions.

Intersection of Graphs

The intersection of graphs represents points where two functions have equal values. To find these intersections algebraically, we equate the given functions. In this case, \( y_1 = \sqrt[3]{x} \) is set equal to \( y_2 = x^2 \). Solving the resulting equation helps identify shared points on both graphs.

• Initially, the equation \( \sqrt[3]{x} = x^2 \) is analyzed.
• We eliminate the cube root by cubing both sides, leading to \( x = x^6 \).
• Reorganization simplifies it to \( x(1 - x^5) = 0 \), giving potential solutions \( x = 0 \) and solving \( 1 - x^5 = 0 \) shows \( x = 1 \).

These points are verified by substitution back into the original functions; they confirm no extraneous solutions exist. Graphically, intersection validates these solutions as where the two visual curves overlap.