Cube Root Function

Cube root of a number is denoted as f(x) = ∛x or f(x) = x^1/3, where x is any real number. It is a number which, when raised to the power of 3, equals to x. The cube root function is the inverse of the cubic function f(x) = x³. A cube root function is a one-one and onto function.

In this article, we will learn about the meaning of the Cube root function, differentiation, and integration of the cube root function, domain and range of the cube root function, properties of cube root functions, and graphing cube root function.

What is Cube Root Function?

Cube root function, denoted as f(x) = ∛x or f(x) = x^1/3, is the inverse of the cubic function f(x) = x³. The cubic function is increasing, one-to-one, and onto, making it a bijection. Consequently, its inverse function, the cube root function, is also a bijection. It returns the unique value such that when raised to the power of 3, it yields the input value (x).

In other words, for every real number x, the cube root function returns the unique real number y such that y³ = x. It is an odd function and has a continuous and smooth graph over its entire domain.

The general form of the cube root function is f(x) = a ∛(bx - h) + k. In this equation, h is the horizontal shift and k is the vertical shift.

Cube Root Function Definition

Cube root of a number (x) is a mathematical function denoted as f(x) = ∛x , where x is any real number. It returns the unique real number y such that y³ = x.

In simpler terms, it is the number which, when multiplied by itself twice, results in (x). The cube root function is the inverse of the cubic function f(x) = x³.

Differentiation of Cube Root Function

Differentiation of the cube root function f(x) = ∛x, can be done by using power rule for differentiation.

The cube root function can also be expressed as f(x) = x^1/3

Using the power rule, the derivative of f(x) with respect to x is:

f'(x) = \frac{d}{dx}(x^{\frac{1}{3}}) = \frac{1}{3}x^{\frac{1}{3} - 1} = \frac{1}{3}x^{-\frac{2}{3}}

So, the derivative of the cube root function is f'(x) = \frac{1}{3x^{\frac{2}{3}}}

Integration of Cube Root Function

To integrate the cube root function (f(x) = ∛x), we can use the power rule of integration.

The cube root function can also be expressed as f(x)= x^1/3.

Integrating f(x) with respect to x gives us:

\int \sqrt[3]{x} \, dx = \int x^{\frac{1}{3}} \, dx

Using the power rule for integration, we add 1 to the exponent and divide by the new exponent:

⇒ 3/4x^4/3 + C

So, the integral of the cube root function is (3/4x^4/3 + C), where (C) is the constant of integration.

Domain and Range of Cube Root Function

Domain of a function refers to all possible input values or (x) values for which the function is defined. In the case of the cube root function f(x) = ∛x, every real number can be plugged in as (x), meaning there are no restrictions on the input values. Therefore, the domain of the cube root function is all real numbers.

Range of a function represents all possible output values or (y) values that the function can produce for the given domain. For the cube root function f(x) = ∛x, when any real number (x) is input, the function yields a unique cube root (y). Since the cube root of any real number is also a real number, the function's outputs cover all real numbers as well. Hence, the range of the cube root function is also all real numbers.

Hence, the cube root function f(x) is \R\to\R: i.e. Domain and range of a cube root function is \R (set of all real numbers)

Asymptotes of Cube Root Function

Concept of asymptotes in functions relates to lines that the graph approaches but does not intersect. Considering the parent cube root function f(x) = ∛x, the following observations can be made:

• As (x) approaches positive or negative infinity (x → ± ∞), f(x) also tends to positive or negative infinity. Consequently, there are no horizontal asymptotes as the function's graph does not approach any horizontal lines without intersecting them.
• Since the cube root function is defined for all real numbers, there are no values of (x) where f(x) is undefined. Thus, there are no vertical asymptotes because the function's graph does not approach any vertical lines without intersecting them.

Hence, it can be concluded that the cube root function does not exhibit any asymptotes.

Graphing Cube Root Functions

To graph cube root functions, follow these steps:

• Determine the Parent Function: The parent cube root function is f(x) = ∛x​. This function starts at the origin (0,0) and extends smoothly in both positive and negative directions.
• Identify Transformations: If there are any transformations applied to the parent function, such as translations or dilations, determine their effects on the graph. Parameters a, h, and k in the general form f(x) = a ∛(x​−h) + k alter the position, size, and orientation of the graph.
• Plot Key Points: Choose several x-values and calculate the corresponding y-values using the function. Plot these points on the coordinate plane.
• Draw the Curve: Connect the plotted points smoothly to form the graph of the cube root function. Ensure that the curve extends smoothly in both directions, reflecting the increasing nature of the function.
• Label Axes and Points: Label the x-axis and y-axis, and mark any significant points on the graph, such as intercepts or points of inflection.

Example: Plot a graph for cube root function g(x) = 4∛(x-3) - 2.

Solution:

Step 1: Graph the parent function i.e. f(x) = ∛x

Below is the graph of f(x) = ∛x. 

Step 2: Change the x and y coordinates as per the function given and plot the new points:

• The new x-coordinates are obtained by equating x - 3 = old x-coordinate, For example: x - 3 = -8 ⇒ x = (-8 + 3) = -5
• The new y-coordinates are obtained by putting ∛x-3 = old y-coordinate in the equation g(x) = 4∛(x-3) - 2. For example: for old y-coordinate = -2 , new y-coordinate = 4(-2) -2 = -10

Below is the graph of g(x) = 4∛(x-3) - 2.

Properties of Cube Root Function

The properties of cube root function are:

• The cube root function is positive on (0, ∞) and negative on (-∞ , 0).
• The domain of the cube root function includes all real numbers. It can accept any real number (x) as input without any restrictions.
• Similarly, the range of the cube root function consists of all real numbers. For any real number (x), there exists a unique cube root (y), making the output range cover all real numbers.
• The cube root function is an odd function, which means that f(-x) = -f(x) for all (x) in its domain. This property reflects symmetry about the origin, where the function's graph is rotationally symmetric by 180 degrees.
• Cube root function does not have any asymptotes.

Cube Root Function vs Square Root Function

The difference between cube root function and square root function is show below in the table:

Applications of the Cube Root Function

Various applications of cube root functions are listed below:

• Engineering: Using in various engineering fields, including fluid dynamics, structural analysis, and electrical engineering, where cubic relationships arise.
• Economics: Applied in economic models to represent scenarios involving cubic growth or decay, such as population growth or depreciation.
• Physics: Used in physics equations to model phenomena with cubic relationships, such as the volume of a cube or the force of an object.
• Computer Graphics: Employed in computer graphics for generating smooth curves and surfaces, particularly in 3D modeling and animation software.
• Mathematics: Integral in calculus for solving cubic equations and analyzing cubic functions, providing insights into complex mathematical problems and modeling real-world phenomena.

Read More,

• Perfect Cubes
• How to find square Root of a number
• Cube Root Calculator

Solved Examples of Cube Root Function

Example 1: Find the domain and range of the function f(x) = ∛(x² - 9)​. Does it have any asymptotes?

Solution:

To find the domain and range of the function f(x) = ∛(x² - 9), we need to determine the values of ( x ) for which the function is defined and the corresponding output values.

1. Domain:

Function is defined for all real numbers ( x ) such that the expression under the cube root, (x² - 9), is non-negative. This means:

x² − 9 ≥ 0

x² ≥ 9

|x | ≥ 3

So, the domain of the function is x ≥ 3 and x ≤ −3, or in interval notation, (-∞, -3) ∪ (3, ∞).

2. Range:

Cube root function will produce real output for all real inputs. Since the cube root of any real number is defined, the range of the function is all real numbers.

3. Asymptotes:

Cube root function does not have any asymptotes because it is continuous and defined for all real numbers.

Therefore, the domain is (-∞, -3) ∪ (3, ∞), the range is all real numbers, and there are no asymptotes for the function f(x) = ∛(x² - 9).

Example 2: Graph the function g(x) = ∛(x − 2) ​- 1 using transformations.

Solution:

To graph the function g(x) = ∛(x−2) ​- 1 using transformations, follow these steps:

1. Start with the parent function f(x) = ∛x​, which represents the cube root function.
2. Apply the transformation f(x−2) to shift the graph of f(x) horizontally right by 2 units.
3. Apply the transformation f(x) - 1 to shift the graph of f(x−2) vertically downward by 1 unit.

Now, let's apply these transformations to the graph:

• The horizontal shift right by 2 units will move the entire graph to the right.
• The vertical shift downward by 1 unit will lower the entire graph.

Graph of the function g(x) = ∛(x − 2) ​- 1 is shown below:

Example 3: Which of the following functions represent cube root functions?

(a) f(x) = ∛x​

(b) f(x) = √x³​

(c) f(x) = x^1/3

(d) f(x) = 1/∛x

Solution:

(a) f(x)=∛x​ This function represents the cube root of x, which is indeed a cube root function.

(b) f(x)=x³​ This function represents the square root of x³, not the cube root. It's not a cube root function.

(c) f(x)=x^1/3 This function represents x raised to the power of 1/3​, which is equivalent to the cube root of x. It's a cube root function.

(d) f(x)=1/∛x This function represents the reciprocal of the cube root of x, not the cube root itself. It's not a cube root function.

So, the cube root functions are options (a) and (c).

Practice Questions on Cube Root Function

Q1. Given the cube root function f(x) = ∛x, find f(8).

Q2. Graph the cube root function f(x) = ∛x without any transformations.

Q3. Determine the domain and range of the function f(x) = ∛x-2.

Q4. If g(x) = 2∛x, what transformation has been applied to the parent cube root function f(x) = ∛x?

Q5. Solve the equation ∛x = 5 to find the value of ( x ).