Problem 93 - Algebra A Combined Function

Question

Identify the domain and then graph each function. $$ g(x)=\sqrt[3]{x-1} $$

Step-by-Step Solution

Answer

The domain is all real numbers: \((-\infty, \infty)\). Graph by shifting the cube root function 1 unit to the right.

Step 1: Understand the Cube Root Function

The function given is \( g(x) = \sqrt[3]{x - 1} \). This is a cube root function. Unlike square roots, cube roots can be evaluated for both negative and positive numbers. Therefore, there are no restrictions on the value of \( x \) due to the cube root.

Step 2: Identify the Domain

Since the cube root function \( \sqrt[3]{x - 1} \) can be evaluated for any real number \( x \), the domain is all real numbers. There are no restrictions based on the expression inside the cube root. Therefore, the domain of \( g(x) \) is \( (-\infty, \infty) \).

Step 3: Graph the Function

To graph \( g(x) = \sqrt[3]{x - 1} \), consider transforming the parent function \( y = \sqrt[3]{x} \). The graph is shifted 1 unit to the right, due to the \( x-1 \) inside the cube root. Plotting a few points might help: \( g(0) = \sqrt[3]{-1} = -1 \), \( g(1) = \sqrt[3]{0} = 0 \), \( g(2) = \sqrt[3]{1} = 1 \), and continue to draw the smooth curve through these points that reflects the nature of a cube root function.

Key Concepts

Domain of a Function: Understanding the Basics

The domain of a function is a fundamental concept in mathematics that defines the set of input values that the function can accept. In the case of a cube root function, such as \( g(x) = \sqrt[3]{x - 1} \), the domain is particularly straightforward.

Cube roots differ from square roots in a critical way. While square roots are only defined for non-negative numbers, cube roots can be calculated for any real number. This property arises because cubing a number (whether positive, negative, or zero) always results in a real number. Consequently, the cube root undoes this operation without any limitations.

For the function \( g(x) = \sqrt[3]{x - 1} \), the expression \( x - 1 \) can be any real number. Thus, there are no restrictions, and the domain of this function is all real numbers, expressed as \( (-\infty, \infty) \). Understanding this concept helps in effortlessly determining domains for similar functions.

Function Transformations: Shifting and Graphing

Function transformations allow us to modify the graph of a function in various ways, such as shifting, stretching, or reflecting it. Specifically, when working with the cube root function like \( g(x) = \sqrt[3]{x - 1} \), understanding these transformations is crucial for accurately sketching its graph.

For \( g(x) = \sqrt[3]{x - 1} \), the transformation involved is a horizontal shift. This occurs because of the \( -1 \) inside the cube root. It implies that every point on the fundamental graph of \( y = \sqrt[3]{x} \) is moved 1 unit to the right.

To visualize this:

• The point on \( y = \sqrt[3]{x} \) where \( x = 0 \) (which is the origin) will shift to \( x = 1 \) on the graph of \( g(x) \).
• Similarly, points such as \( (1, 1) \) on \( y = \sqrt[3]{x} \) move to \( (2, 1) \) on the graph of \( g(x) \).

These transformations are essential to understand, as they allow not only a graphical representation but also provide insight into how the function behaves when the input is altered. Graphed effectively, these transformations allow us to create a curve that reflects the smooth and continuous nature of cube root functions.

Real Numbers and Their Role in Mathematics

Real numbers encompass a comprehensive range of values that include all the numbers you can think of: positive numbers, negative numbers, zero, and even fractions and decimals. They are central not only to the cube root function but to most mathematical concepts.

In the context of cube root functions like \( g(x) = \sqrt[3]{x - 1} \), real numbers define both the domain and the range. Since real numbers include negatives, it is possible to take a cube root of any real number, which gives the function an unrestricted domain. This is why the domain of \( g(x) \) is \( (-\infty, \infty) \).

Importantly, the idea of real numbers is also linked closely to continuous graphs. Functions dealing with real numbers tend to have smooth, unbroken curves as their graphs, as is the case with the cube root function. Understanding real numbers ensures a deep comprehension of continuity, limits, and other fundamental concepts in calculus and algebra, thus enhancing problem-solving abilities across various mathematical disciplines.