Problem 91 - Intermediate Algebra

Question

Identify the domain and then graph each function. \(g(x)=\sqrt[3]{x-1} ;\) use the following table. $$ \begin{array}{|c|c|} \hline x & {g(x)} \\ \hline 1 & {} \\ \hline 2 & {} \\ \hline 0 & {} \\ \hline 9 \\ \hline-7 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The domain is all real numbers, and the graph follows an S-curve path, increasing throughout, with points (1,0), (2,1), (0,-1), (9,2), (-7,-2).

Step 1: Identify the Domain

To find the domain of the function \(g(x) = \sqrt[3]{x-1}\), we need to understand that for a cube root, the domain is all real numbers. This is because the cube root function can accept any real number input, and there is no restriction on subtracting 1 from \(x\). Therefore, the domain of the function is all real numbers, \(x \in \mathbb{R}\).

Step 2: Calculate \(g(x)\) for Specific \(x\) Values

Using the function \(g(x) = \sqrt[3]{x-1}\), we will evaluate \(g(x)\) for each \(x\) value in the table.\- For \(x = 1\), \(g(1) = \sqrt[3]{1-1} = \sqrt[3]{0} = 0.\)\- For \(x = 2\), \(g(2) = \sqrt[3]{2-1} = \sqrt[3]{1} = 1.\)\- For \(x = 0\), \(g(0) = \sqrt[3]{0-1} = \sqrt[3]{-1} = -1.\)\- For \(x = 9\), \(g(9) = \sqrt[3]{9-1} = \sqrt[3]{8} = 2.\)\- For \(x = -7\), \(g(-7) = \sqrt[3]{-7-1} = \sqrt[3]{-8} = -2.\)

Step 3: Graph the Function

Plot the calculated points from the table onto a graph. These points are (1, 0), (2, 1), (0, -1), (9, 2), and (-7, -2). Since the function is a cube root, the general shape of the graph will be an S-curve passing through the origin. The graph will extend infinitely in both directions, reflecting the domain of all real numbers.

Step 4: Analyze the Graph's Behavior

The graph of \(g(x) = \sqrt[3]{x-1}\) shows that it is an increasing function. As \(x\) increases, \(g(x)\) increases, which is typical for cube root functions. There is no asymptotic or bounded behavior. The graph's form resembles an S-curve, demonstrating smooth and continuous growth.

Key Concepts

Domain of a Function

In mathematics, the domain of a function involves identifying all possible input values, usually denoted as \(x\), that result in a valid output. Without understanding the domain, it would be like operating a machine without knowing all the controls. For the function \(g(x) = \sqrt[3]{x-1}\), the domain encompasses every real number. This is because for cube root functions, you can plug in any real number, and you'll get a valid output. To imagine this, think about a set of endless boxes where any number can fit. No matter how large or small the number, or whether it is positive or negative, a cube root function embraces it all. In mathematical terms, this means the domain is all real numbers,

• \(x \in \mathbb{R}\)

You do not have to worry about restrictions like dividing by zero or dealing with negatives inside even roots, which is refreshing! This quality makes cube root functions particularly easy to work with.

Graphing Functions

Graphing a function is like painting its life story on a canvas. For the function \(g(x)=\sqrt[3]{x-1}\), graphing presents a unique shape called an S-curve. To construct the graph, you begin by plotting key points from a table. In our specific example, the points are determined by substituting select \(x\) values into the cube root formula. These calculated points, such as (1, 0) and (2, 1), provide the bones of your graph. Once plotted, the visualization starts to form as you connect the dots.

• Points to plot: (1, 0), (2, 1), (0, -1), (9, 2), (-7, -2).

When graphed, the curve reveals itself as a continuous and smooth S-shape. As you gaze at this S-curve, note how it spans indefinitely left and right, indicating the function’s vast domain. This flow visually confirms each real number \(x\) corresponds to a \(g(x)\), showing the ease and elegance of cube root functions.
While graphing, there's immense power in visualization. It makes abstract math tangible, helping to better grasp both growth patterns and behavior of functions.

Real Numbers

Real numbers are the broad classification covering almost every figure you can think of when you conjure up numbers, both rational and irrational. This group holds integers like -1 or 3, fractions like 1/2, and irregular numbers like \(\sqrt{2}\) or \(\pi\). They exist on the number line and are used to plot functions like our cube root example.

• Real numbers include both fractions and whole numbers.
• They cover both rational numbers and irrational numbers, which cannot be precisely expressed as fractions.

Understanding real numbers is crucial because they inform us about what we can input into functions and what we expect in outputs. For \(g(x) = \sqrt[3]{x-1}\), every real number from minus infinity to infinity works without a hitch, and it is a reassuring fact that tells us the cube root gently embraces the entire world of real numbers without prejudice!
Comprehending the essence of real numbers reassures us there's a universe of data points ready to be explored along a graph. The concept bridges the abstract with the actual, reflecting how mathematics beautifully intertwines with reality.