Problem 85 - College Algebra with Corequisite Support

Question

For the following exercises, graph \(y=\sqrt[3]{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. [-0.001,0.001]

Step-by-Step Solution

Answer

The range for \([-0.001, 0.001]\) is \([-0.1, 0.1]\).

Step 1: Understand the Function

To begin, we need to understand what the function \(y = \sqrt[3]{x}\) represents. A cube root function transforms the input \(x\) into its cube root. The graph of this function has a characteristic S-shape, with the key point at the origin (0,0) where the input is zero.

Step 2: Identify the Viewing Window

The given viewing window is from \([-0.001, 0.001]\). This range is very narrow, centered around the origin. It is important to consider that this range includes both negative, positive, and zero values of \(x\), which allows us to capture both the decreasing and increasing parts of the function near the origin.

Step 3: Determine the Range

For the cube root function, all real numbers \(x\) have corresponding real \(y\) values. Specifically, since \([-0.001, 0.001]\) is our domain, the range will also be \([-\sqrt[3]{0.001}, \sqrt[3]{0.001}]\). Computing the cube root, we find: \(\sqrt[3]{0.001} = 0.1\), so the range is \([-0.1, 0.1]\).

Step 4: Graph the Function

On the graph, plot \(y = \sqrt[3]{x}\) using the viewing window \([-0.001, 0.001]\) on the x-axis, which will reflect the range \([-0.1, 0.1]\) on the y-axis. This will show the characteristic subtle curve through the origin, with the graph smoothly transitioning through the tiny range.

Key Concepts

Cube Root Function

The cube root function, expressed as \(y = \sqrt[3]{x}\), is a type of radical function where each value of \(x\) is transformed into its cube root. Unlike square root functions, cube root functions can accept negative inputs since the cube of any real number always results in a real number. The defining shape of the graph of a cube root function is the S-curve. This gentle curve balances itself symmetrically around the origin \((0,0)\), moving smoothly from negative infinity through zero and up towards positive infinity.

Key features of the cube root function include:

• Continuity across all real numbers. There are no breaks or undefined points.
• Its domain and range extend over all real numbers.
• Symmetry about the origin, showcasing odd function characteristics.

By understanding these properties, you'll find it easier to predict and interpret the behavior of cube root graphs across different domains.

Graphing Techniques

Graphing the cube root function involves using specific techniques to accurately capture its S-shaped curve. Starting with the basic understanding of the function's shape helps in plotting it over any specified interval effectively. For the cube root function \(y = \sqrt[3]{x}\):

• Identify the domain: In this exercise, the domain is limited to a narrow window \([-0.001, 0.001]\).
• Locate key points: For cube root functions, crucial points are where \(x = 0\), \(x = 1\), and \(x = -1\).
• Sketch the curve: Plot values and use symmetry about the origin to ensure the curve accurately reflects the gradual shift from decreasing to increasing values.

The tight viewing window \([-0.001, 0.001]\) challenges us to closely observe how the curve behaves around the origin. This interval allows us to focus on the subtle transition seen in the infinitesimally small changes in \(x\) and their corresponding \(y\) values, providing a microscopic view of the function's nature.

Range of a Function

The range of a function encompasses all the possible values it can output as \(y\) for any given \(x\) within the domain. For the cube root function \(y = \sqrt[3]{x}\), since it maps every real number \(x\) to a real number \(y\), theoretically, its range is all real numbers. However, when constrained within a specific domain like \([-0.001, 0.001]\), the range is also confined.

In this case:

• Compute the extrema of \(y\), determined by \(x = -0.001\) and \(x = 0.001\).
• Calculate the cube roots \([-\sqrt[3]{0.001}, \sqrt[3]{0.001}]\).
• Result: Range is \([-0.1, 0.1]\)

This small window showcases a detailed portion of the S-curve, allowing insights into how \(y = \sqrt[3]{x}\) transitions smoothly through this tiny interval. Understanding the concept behind the range of a function helps to predict possible output values even when restricted, enhancing one's ability to interpret function behavior in context.