Domain Of Cube Root Function Y = \sqrt[3]{x-1} + 3: A Comprehensive Guide

Jul 10, 2025 by Admin 74 views

$Exploring the Domain of the Translated Cube Root Function y = \sqrt[3]{x-1} + 3$

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The domain of a function is a fundamental concept in mathematics that defines the set of all possible input values (x-values) for which the function produces a valid output (y-value). In simpler terms, it's the range of x-values you're allowed to plug into the function without causing any mathematical errors, such as dividing by zero or taking the square root of a negative number. Understanding the domain is crucial for grasping the behavior and limitations of a function. In this article, we delve into the domain of a specific function: ${ y = \sqrt[3]{x-1} + 3 }$ . This function represents a cube root transformation, a concept we'll explore in detail.

Understanding Cube Root Functions

Before we tackle the specific function at hand, let's first understand the nature of cube root functions in general. A cube root function is a function of the form ${ f(x) = \sqrt[3]{x} }$ , where the cube root symbol indicates the inverse operation of cubing a number. Unlike square roots, which only accept non-negative inputs, cube roots can handle both positive and negative numbers, as well as zero. This is because any real number, whether positive, negative, or zero, has a unique cube root.

For example, the cube root of 8 is 2 because ${ 2^3 = 8 }$ . The cube root of -8 is -2 because ${ (-2)^3 = -8 }$ . And the cube root of 0 is 0 because ${ 0^3 = 0 }$ . This property of cube roots is key to understanding why the domain of the basic cube root function ${ y = \sqrt[3]{x} }$ is all real numbers. There are no restrictions on the values we can input into the cube root function.

Transformations of Cube Root Functions

Our function of interest, ${ y = \sqrt[3]{x-1} + 3 }$ , is not just a basic cube root function; it's a transformation of the basic cube root function ${ y = \sqrt[3]{x} }$ . Transformations involve altering the graph of a function by shifting, stretching, compressing, or reflecting it. In this case, we have two transformations at play:

Horizontal Translation: The term ${ (x - 1) }$ inside the cube root shifts the graph horizontally. Specifically, it shifts the graph 1 unit to the right. This is because to get the same y-value as in the original function, we need to input a value that is 1 greater. For example, to get the same y-value when x = 0 in ${ y = \sqrt[3]{x} }$ , we need to input x = 1 in ${ y = \sqrt[3]{x-1} }$ .
Vertical Translation: The addition of 3 outside the cube root shifts the graph vertically. In this case, it shifts the graph 3 units upward. This is a straightforward vertical shift; every y-value is increased by 3.

These transformations affect the position of the graph in the coordinate plane but do not fundamentally change the nature of the cube root function. The graph still extends infinitely in both the positive and negative x-directions. Therefore, the transformations do not introduce any new restrictions on the domain.

Determining the Domain of y = \sqrt[3]{x-1} + 3

Now, let's get to the heart of the matter: determining the domain of the function ${ y = \sqrt[3]{x-1} + 3 }$ . As we discussed earlier, the cube root function itself does not impose any restrictions on the domain. We can take the cube root of any real number.

The horizontal translation, represented by the ${ (x - 1) }$ term, simply shifts the graph left or right. It doesn't eliminate any x-values from the domain. Similarly, the vertical translation, represented by the ${ + 3 }$ term, shifts the graph up or down and does not affect the domain.

Therefore, the domain of ${ y = \sqrt[3]{x-1} + 3 }$ is the same as the domain of the basic cube root function ${ y = \sqrt[3]{x} }$ : all real numbers. This means we can input any real number into the function, and it will produce a valid real number output.

Expressing the Domain

In mathematical notation, we can express the domain of ${ y = \sqrt[3]{x-1} + 3 }$ in several ways:

Set Notation: {x | x is a real number}
Interval Notation: ${ (-\infty, \infty) }$

Both notations convey the same meaning: that the domain includes all possible real numbers, from negative infinity to positive infinity.

Analyzing the Given Options

Now, let's examine the answer choices provided in the original problem:

A. {x | 1}: This option suggests that the domain is restricted to x-values greater than 1. This is incorrect because the cube root function is defined for all real numbers, including those less than or equal to 1.
B. {y | 1}: This option refers to the range of the function (the set of possible y-values), not the domain. While the range is also all real numbers in this case, this option is not relevant to the question about the domain.
C. {x | x is a real number }: This is the correct answer. It accurately states that the domain of the function is all real numbers.
D. {y | y is a real number }: Similar to option B, this refers to the range of the function, not the domain.

Therefore, the correct answer is C. {x | x is a real number }.

Visualizing the Domain on the Graph

The graph of ${ y = \sqrt[3]{x-1} + 3 }$ visually confirms our conclusion about the domain. The graph extends infinitely to the left and right along the x-axis, indicating that there are no x-values excluded from the domain. No matter how far left or right you go on the graph, you will always find a corresponding point on the curve.

The horizontal shift of 1 unit to the right and the vertical shift of 3 units upward only change the position of the graph; they do not create any breaks or gaps in the x-values covered by the function.

Conclusion

In conclusion, the domain of the translated cube root function ${ y = \sqrt[3]{x-1} + 3 }$ is all real numbers. This is because the cube root function itself has a domain of all real numbers, and the transformations applied (horizontal and vertical translations) do not introduce any new restrictions on the domain. Understanding the nature of cube root functions and how transformations affect their graphs is essential for accurately determining their domains.

Remember, the domain of a function is a critical aspect of its behavior, defining the set of valid inputs. By carefully analyzing the function's structure and considering any transformations, we can confidently determine its domain and gain a deeper understanding of its properties.

On a coordinate grid, the graph of ${ y = \sqrt[3]{x-1} + 3 }$ , which is a transformation of ${ y = \sqrt[3]{x} }$ , is shown. What is the domain of the graphed function?

Domain of Cube Root Function y = \sqrt[3]{x-1} + 3: A Comprehensive Guide