A cubic function is a polynomial function of degree 3 and is represented as f(x) = ax³ + bx² + cx + d, where a, b, c, and d are real numbers and a ≠ 0. Cubic functions have one or three real roots and always have at least one real root. The basic cubic function is f(x) = x³

Let’s learn more about the Cubic function, its domain and range, asymptotes, intercepts, critical and inflection points, and others along with some detailed examples in this article.

What is Cubic Function?

Any polynomial function of degree 3 is called a cubic function. The cubic function is ax³ + bx² + cx + d, where a, b, c, and d are real numbers and a ≠ 0.

Standard Form of Cubic Function = ax³ + bx² + cx + d

Note: The fundamental cubic function is f(p) = p³.

Domain and Range of Cubic Function

Domain and Range of Function of a function are the values of x(independent variable) for which the function is defined and, all the values of y(dependent variables) that are contained after substituting x to the function, respectively.

Cubic function y = f(x) is defined for the set of all real numbers for x. So, the domain of the cubic function is a set of all real numbers. The value of y obtained also covers the set of all real numbers. So, the range of the cubic function is a set of all real numbers.

• Domain of Cubic Function: (−∞,∞) or R

• Range of Cubic Function: (−∞,∞) or R

Asymptotes of Cube Function

Asymptotes are referred to as the values that are not in the domain and range. The domain and range of the cubic function includes the set of all real numbers which implies that there is no asymptote for cubic functions.

So, the cubic function does not have a vertical asymptote or horizontal asymptote.

Roots of Cubic Function

Roots of a cubic function, also known as its zeros or x-intercepts, are the values of x where the function f(x) equals zero.

General form of a cubic function is ax³ + bx² + cx + d, where a, b, c, and d are constants, and a is not equal to zero.

Factors of Cubic Function

Factors are those expressions which can divide the cubic function exactly without leaving any remainder.

A cubic function has the highest degree ‘3’, hence it only has at most three factors. If m, n and o are the root of the cubic function then the cubic function can be written as:

a(x – m)(x – n)(x – o) = 0

where a is the Leading Coefficient.

Here, (x – m), (x – n), and (x – o) are three factors of cubic function.

Read More

• Factorization of Polynomial

How to Solve Cubic Functions?

To solve any cubic equation follow the steps added below:

Step 1: Start with the cubic function f(x) = ax³ + bx² + cx + d.

Step 2: Set the function equal to zero: f(x) = 0.

Step 3: Use one of the following methods to find the roots:

Step 4: Factorize if possible.

Step 5: Check for complex roots if necessary.

Step 6: Verify solutions by plugging them back into the original equation.

Express solutions in the desired form (e.g., decimal approximation or exact form).

Intercepts of a Cubic Function

Intercepts of a cubic function are the points where the graph of the function intersects the x-axis and the y-axis. There can be two intercepts for any function based on the intersection with the axis i.e.,

• x-Intercept
• y-Intercept

X-Intercept

x-intercept of cubic function is obtained by solving the function by putting f(x) = 0. We know that the degree of the cubic function is 3 so, there are a maximum of 3 roots of the cubic function.

Thus, there can be a maximum of three x-intercepts for any cubic function.

Y-Intercept

y-intercept of the cubic function is obtained by by putting x = 0 in the function and determining the value of f(x) [i.e., y]. There is exactly one y-intercept for a cubic function.

For example, consider the cubic function f(x)= ax³ + bx² + cx + d.

• To find x-intercepts, set f(x) = 0 and solve the cubic equation ax³ + bx² + cx + d = 0 for x.

• To find y-intercept, substitute x = 0 into the function f(x) to get f(0) i.e., d.

Read More

• X and Y Intercepts

Graph of Cubic Function

A cubic function is typically represented by the equation ax³ + bx² + cx + d., where a, b, c, and d are constants. Depending on the specific values of these constants, the graph of a cubic function can take various shapes. Graph of the cubic function exhibits S shape graph.

Graph of fundamental cubic function i.e., f(x) = x³ is given as follows:

Read More

• Graph of Polynomial

Characteristics of Cubic Function

There are multiple characteristics of a cubic function some of these are listed below:

• A cubic function has degree 3.
• It can have one or three real roots.
• It can have zero or two complex roots.
• It has only one inflection point.
• A cubic function can have two critical points local maxima and local minima.
• Cubic functions may exhibit symmetry depending on the coefficients.
  • They can be odd functions (symmetric about the origin) if b = 0
  • They can be even functions (symmetric about the y-axis) if c = 0.

Inverse of Cubic Function

Steps to find the inverse of the cubic function are:

Step 1: Write the cubic function as y = f(x).

Step 2: Swap the places of ‘x’ and ‘y’

Step 3: Simplify and solve for ‘y’ if possible.

Step 4: Obtained value of y is the inverse of the cubic function.

Inverse of the cube function is the cube root function i.e., for f(x) = x³,

f^-1(x) = ∛x

For Example: Find the inverse of the cubic function f(x) = x³ + 2.

Solution:

Let y = f(x) = x³ + 2

Now, Swap the places of ‘x’ and ‘y’

x = y³ + 2

⇒ y³ = x – 2

⇒ y = ∛(x – 2)

Since y = f^-1(x)

f^-1(x) = ∛(x – 2)

So, the inverse of given cubic function is ∛(x – 2).

Extrema of Cubic Function

Extrema refers to the maximum or minimum value of a function within a given interval or over its entire domain. For any polynomial function of degree n there can be at most n – 1 extrema.

Note: If the cubic function is always increasing or always decreasing over its entire domain, it does not have any local extrema.

To find the extrema of any function we need to find critical points and inflection points for some cases as well. Let’s discuss both in detail with the help of some examples as well.

Critical Points of Cubic Function

The critical point is referred to as the point where the function has its minimum or maximum value. In other words, the point where the function changes its behavior from increasing to decreasing or vice-versa then, the point is called as critical point of the function. To find the critical points of cubic function we equate the first-order derivative of the function with zero and then solve the resultant equation.

For example, consider the cubic function f (x) = ax³ + bx² + cx + d.

Steps to Find Critical Point of a Function

Step 1: Determine f'(x) for, given f(x) i.e., f'(x) = 3ax² + 2bx + c

Step 2: Equate f'(x) = 0 i.e., 3ax² + 2bx + c = 0

Step 3: Solve for x

Using Quadratic Formula

x = {-2b ± √(4b² – 12ac)}/6a

x = {-b ± √(b² – 3ac)}/3a

Step 4: Values of x are the critical points of the above cubic equation, i.e.

• f(x) has Two Critical Points if b² – 3ac > 0
• f(x) has One Critical Point if b² – 3ac = 0
• f(x) has No Critical Point if b² – 3ac < 0

Inflection Points of Cubic Function

Inflection points are referred to as the points where the function curve changes from concave up to concave down or vice-versa. To find the inflection points of the cubic function we equate the second-order derivative of the cubic function with zero and solve.

For example, consider the cubic function f (x) = ax³ + bx² + cx + d.

Steps to Find Inflection Points of Cubic Function

Step 1: Determine f”(x) of given function f(x) i.e., f”(x) = 6ax + 2b

Step 2: Equate f”(x) = 0 i.e., 6ax+ 2b = 0

Step 3: Solve for x we get x = -2b/ 6a

Step 4: Value of x = -b /3a is the inflection point of the above cubic equation.

Thus, cubic function f(x) = ax³ + bx² + cx + d has inflection point at {-b/3a, f(-b/3a)}.

End Behavior of Cube Function

We know that the cubic function is of the form ax³ + bx² + cx + d where ‘a’ is called the leading coefficient. Based on the leading coefficient we can obtain the end behavior of the cube function. There are two cases for the end behaviour of the cubic function.

• When Leading Coefficient is Positive

• When Leading Coefficient is Negative

When Leading Coefficient is Positive

When the leading coefficient is positive i.e., a > 0 then, the cubic function depicts the following behaviour.

f(x) → ∞ as x → ∞

f(x) → –∞ as x → –∞

When Leading Coefficient is Negative

When the leading coefficient is negative i.e., a < 0 then, the cubic function depicts the following behaviour.

f(x) → –∞ as x → ∞

f(x) → ∞ as x → –∞

Graphing Cubic Function

To graph a cubic function {f(x) = ax³ – bx² + cx + d} follow the steps added below:

Step 1: Find x-intercept(s)

Step 2: Find y-intercept

Step 3: Find critical point(s) by setting f'(x) = 0

Step 4: Find the corresponding y-coordinate(s) of the critical points.

Step 5: Find end behavior of the function.

Step 6: Plot all the points obtained, and trace the required cubic function.

Cubic Function Vs Quadratic Function

Key differences between cubic and quadratic functions are: