It is important to know the classification of functions in order to learn the major concepts in mathematics, particularly for U.S. competitive tests such as the SAT, ACT, and GRE. Classification of functions makes it simple to recognize and examine various functions in terms of their expressions, domain, and range. The domain may be numbers, decimals, angles, integers, or fractions, but the range, represented by p or f(q), is usually made up of numerical values. An operation, defined as p = f(q), is classified according to these aspects. As functions are pivotal in solving algebraic equations and describing real-life situations, an excellent understanding of these categorizations can provide you with a major advantage in cracking competitive exams.

What are Types of Functions?

Types of functions in math are determined based on the domain, range and function expression. The expression applied to address the function is the principal defining factor for a function. Types of functions have been grouped into the below types:

• Based on set elements
• Based on equation
• Based on range
• Based on domain

Various types of functions are mentioned in the below table:

Let us learn about each of the above-listed functions with examples and diagrams.

Types of Functions based on Set Elements

Types of functions based on set elements depend on the number of relationships amongst the elements in the domain and the codomain. The different types of functions depending on the set elements are as discussed below.

1. One–One Function or Injective Function

The one-to-one function is also termed an injective function. Here each element of the domain possesses a different image or co-domain element for the assigned function.

A function f: A → B is declared to be a one-one function if different components in A have different images or are associated with different elements in B.

2. Onto Function or Surjective Function

A function f: A → B is declared to be an onto function if each component in B has at least one pre-image in A. i.e., If-Range of function f = Co-domain of function f, then f is onto. The onto function is also termed a subjective function.

3. Bijective Function or One One and Onto Function

A function f: A → B is declared to be a bijective function if it is both one-one and onto function. In other words, we can say that every element of set A is related to a different element in set B, and there is not a single element in set B that has been left out to be connected to set A.

4. Many-One Function

Any function f: A → B is said to be many-one if two (or more than two) distinct components in A have identical images in B. In a many-to-one function, more than one element owns the same co-domain or image.

5. Into Function

Any function f: A → B is said to be an into function if there exists at least one element in B which does not have a pre-image in A. i.e., If the Range of function f ⊂ Co-domain of function f, then f is into.

To summarise we can say that the into function is precisely opposite in features to an onto function. That is here certain elements in the co-domain do not own any pre-image. This states that the elements in set B are excess and are not equated to any elements in set A.

6. Constant Function

A constant function is a significant form of a many-to-one function. In this function, all the domain elements possess a single data/ image.

A constant function is the sort of function that presents the same value of output for any presented input. It is represented as, f(x) = c, where c is a constant. For example, f(x) = 6 is a constant function.

OR

The constant function is mathematically expressed as f: R→R and is represented as f(x) = y = c, for x ∈ R and c denotes a constant in R. The domain of the function f signifies R and its range is a constant, c. Sketching a graph, we obtain a straight line that is parallel to the x-axis as displayed above.

Types of Function based on Equation

Algebraic expressions are also included in the types of functions and are based on the degree of the polynomial expression. For example:

• The polynomial function with degree zero is declared to be a constant function.
• The polynomial function of degree one is termed a linear function.
• The polynomial function of degree two is termed a quadratic function.
• Similarly, the polynomial function of degree three is a cubic function.

Identity Function

The identity function is the kind of function that provides an identical input as the output. It is represented as, f(x) = x, where x ∈ R. For example, f(4) = 4 denotes an identity function. This implies that the identity function possesses an identical domain and range. The domain and range of the identity function are of the pattern {(1, 1), (2, 2), (3, 3), (4, 4)…..(n, n)}.

The graph of the identity function is a straight continuous line that is fairly inclined to the coordinate axes and is crossing through the origin. The identity function can practice both positive and negative values and therefore it is present in the first and the third quadrants of the coordinate axis as can be seen from the above graph.

Linear Function

A polynomial function with the first-degree equation is said to be a linear function. The domain and range for such a function is a real number, and it produces a straight-line graph. Equations such as y = x + 4, y = 6x, y = 4x – 1, are all examples of linear functions. The identity function of y = x can also be included in the linear function.

Check out the graph for y = x – 6.

The general form of a linear function is f(x) = px + q, where p, q are real numbers. Graphically the linear function can be interpreted by the equation of a line y = mx + c, where m denotes the slope of the line and c implies the y-intercept of the line.

Polynomial Function

A Polynomial function is a sort of function that can be represented as a polynomial. It is expressed as, \(f\left(x\right)=a_0+a_1x+a_2x^2+\dots+a_nx^n\), where n represents a non-negative integer and \(a_0,a_1,a_2,\dots,a_n∈R.\). For example, \(f\left(x\right)=3x+7\) is a polynomial function.

Quadratic Function

A Quadratic function is a kind of function that holds the highest power two in the polynomial function. For example, \(f(x)=x^{2}-4\) is a quadratic function. The graph of a quadratic equation follows a non-linear pattern and is parabolic in shape as can be seen from the above graph.

In other words, a quadratic function is one with a second-degree quadratic equation and it has a graph that forms a curve. The general pattern of the quadratic function is \(f(x) = ax^{2} + bx + c\), where a ≠ 0 and a, b, c are constant with x as the variable. The domain and range of the quadratic function is R.

Cubic Function

A cubic function as the name implies is a sort of function that has the highest power three in the polynomial function. For example, \(f(x)=x^3+6\) is a cubic function.

The general form of a cubic function is \(f(x) = ax^{3} + bx^{2} + cx +d\), where a ≠ 0 and a, b, c, and d denote the real numbers and x is a variable.

The graph of a cubic function is more curved than the quadratic function. The domain and the range are R. Check out the graph for \(y = x^{3} – 5\).

Types of Functions based on Range

Inline is the type of function based on the range which is received from the given functions. The different types of functions depending on the range are discussed below.

Modulus Function

The function \(f\left(x\right)=\left|x\right|\)is called a modulus function.

This can be further defined as:

\(\begin{Bmatrix}f\left(x\right)&=x&x\ge0\\
f\left(x\right)&=-x &x<0\end{Bmatrix}\)

The modulus function returns the absolute value of the given function, irrespective of the sign of the input domain content. The graph of a modulus function extends in the first and the second quadrants as the coordinates of the points on the graph are of the pattern (x, y), (-x, y).

The domain of |x| is R and its range is [0, ∞).

The function f: R →R defined by f(x) = |x| for each x ∈R is called the modulus function. This implies that for every non-negative value of x, f(x) is equivalent to x. Although for negative conditions of x, the value of f(x) is negative concerning the value of x.

Rational Functions

A Rational function is a sort of function that is derived from the ratio of two given polynomial functions and is expressed as,\(f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}\), such that P and Q are polynomial functions of x and Q(x) ≠ 0. For example, \(f\left(x\right)=\frac{\left(x^2+4x+5\right)}{x^{2}−4}\)  is a rational function.

Consider the below graph for the rational function given by the equation: \(y=\frac{4x+1}{2x+1}\).

The domain and range of rational functions are R. The graphical illustration shows asymptotes, the curves which appear to touch the axes-lines as can be outlined from the above graph.

Signum Function

Signum function helps determine the sign of the real value function and attributes +1(positive 1) for positive input values of the function, and -1(negative 1) for negative input values of the function.

The signum function simply yields the sign for the assigned values of x. For x value higher than zero, the value assigned to the output is +1, for x value lesser than zero, the value assigned to the output is -1, and for x value equal to zero, the output is equivalent to zero. The signum function can be interpreted and learned from the below expression.

The signum function f: R→R represented by:

\(\begin{matrix}f\left(x\right)&= x&x>0\\
&=\ 0&x=0\\
&=-x& x<0\end{matrix}\)

OR

\(\begin{matrix}f\left(x\right)&=\frac{\left|x\right|}{x}&\text{ if } x\ne0\\
&=0&\text{ if }x=0\end{matrix}\)

The graph for the signum function is shown below:

The domain of the signum function covers all the real numbers and is represented along the x-axis, and the range of the signum function has simply two values, +1, -1, drawn on the y-axis.

Domain = R

Range = {-1, 0, 1}

Even and Odd Function

The even and odd functions depend on the relationship between the input and the output states of the function. That is for the negative domain value, if the range is also a -ve value of the range of the primary function, then the function is said to be an odd one.

Furthermore for a -ve domain value, if the range is equivalent to that of the primary function, then the function denotes an even one.

In terms of mathematical expression; if f(-x) = f(x), for all the values of x, then the function is considered to be an even function, and if f(-x) = -f(x), for all the given values of x, then the function is said to be an odd function. Consider the below even and odd function examples:

\(f(x) = x^{3}sinx\)

\(f(-x) = x^{3}sinx\)

Here, f(-x) = f(x)

Therefore the above function is an even function.

Similarly;\(f(x)={{x}^{3}}\) where \(f(-x)={{-x}^{3}}\)

That is; f(x)=-f(-x).

Therefore the function \(f(x)={{x}^{3}}\) is an odd function.

Inverse Function 

Let f: A → B be one-one and onto (bijective) function. Then \(f^{-1}\)  exists which is a function \(f^{-1}\) denoted by; B → A, which maps every component b ∈ B with a component a ∈ A such that f(a) = b is termed as the inverse function of f: A → B.

For inverse of a function the domain and range of the assigned function are reverted concerning the range and domain of the inverse function. The inverse of a function is prominently observed in algebraic functions and inverse trigonometric functions.

Periodic Function

The function is thought to be a periodic function if the same range sequentially resembles the different domain values. We can say that the trigonometric functions are periodic.

For example, the function f(x) = Sinx, has a range equal to the range of [-1, 1] for the various domain values.

Greatest Integer Function

The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ≤ x.

OR

The function f: R → R represented by f(x) = [x], x∈R assumes the value of the greatest integer, less or equal to x. Such a function is designated as the greatest integer function.

The greatest integer function is also recognized as the step function which can be visualized in the above diagram. The domain for such a function is real numbers R, while its range comprises integers (Z).

Smallest Integer Functions

The function f (x) = [x] is called the least/smallest integer function and means the smallest integer greater than or equal to x i.e [x] ≥ x.

OR

The function f: R → R represented by f(x) = [x], x∈R understands the value of the smallest integer, greater or equal to x. Such a function is designated as the smallest integer function.

Composite Function

The composite functions are of the form of fog(x), gof(x), h(g(f(x))), and are composed of individual functions like f(x), g(x), h(x).

For example:

\(f(x)=x^2\text{ and }g(x)=4x\)

\(f(g(x))=f(4x)=(4x)^2=16x^2\)

 

\(g(f(x))=g(x^2)=4x^2\)

Types of Functions based on Domain

Functions are practiced in many other topics of maths. The function equations usually hold algebraic representations, trigonometric, logarithms and exponents and therefore are named based on these domain values. Check out more about the domain-based functions in this section.

Algebraic Function

A function that involves a finite number of terms including powers, roots of independent variable x, coefficient, constant term, plus fundamental operations like addition, multiplication, subtraction, and division are recognized as an algebraic equation.

\(f(x)=6x^3-2x^2+4x+7,g(x)=\frac{\sqrt{3x+6}}{(x-3)^2}\).

An algebraic function is essential to determine the various operations of algebra and is also identified as a linear function, cubic function, quadratic function, or polynomial function, depending on the degree of the algebraic equation.

Trigonometric Functions

The six basic trigonometric functions are sinθ, cosθ, tanθ, secθ, cosecθ. Hereabouts the domain value θ is the angle and is measured in degrees or radians. These trigonometric functions can be defined through the ratio of the sides of a right-angle triangle, via the Pythagoras theorem.

The trigonometric functions along with the inverse trigonometric functions are also sometimes included in periodic functions as the principal values are repeated after a fixed period of time.

Exponential Functions

An exponential function is mathematically expressed as\(f\left(x\right)=f\left(y\right)=b^y\), where “y” stands for the variable and “b” denotes the constant which is also termed as the base of the function.

The exponential function is mainly employed to determine exponential decay/ exponential growth. The most commonly used exponential function base is e.

Logarithmic Functions

The representation of the Logarithmic functions as shown;\(f\left(y\right)=\log_b\left(y\right)\).

Here b signifies the base of the function.

Depending upon the base the function can be a decreasing(value of b lies between 0 to 1) function or an increasing(value b is greater than 1 ) function. Logarithmic functions are also the inverse of exponential functions.

Representation of Functions

The functions require to be designed to display the domain values and the range values and the relationship or link between them. There are three distinct forms of representation of functions and they are Venn diagrams, graphical forms, and roster patterns. Three of the patterns are discussed below.

Venn Diagram: The Venn diagram is a powerful form for describing the function. The Venn diagrams are normally displayed by two circles with arrows combining the components in each of the circles. The domain is shown in one circle and the range values are placed in another one. Also, the function specifies the arrows, and how the arrows relate to the different elements in the two given circles.

Graphical Form: Functions are straightforward to understand if they are represented in a graphical pattern with the use of the coordinate axes. Expressing the function in the graphical form helps us to learn the changing operation of the functions if the function is progressing or declining.

The domain of the function that is the p-value is represented on the x-axis, and the range or the f(p) state of the function is plotted on the y-axis respectively.

Roster Form: Roster notation or the roster form of a set is a simple mathematical representation of the set. The domain and range of the function are expressed in brackets with the first component of a pair denoting the domain and the second component expressing the range.

Let us try to surmise this with the help of a simple example. For a function of the pattern \(f(x) = x^{3}\), the function is represented as {(1, 1), (2, 8), (3, 27), (4, 64)}. Herein the first element denotes the domain or the x value and the second component signifies the range or the f(x) value of the function.

Solved Examples of Types of Functions

With the knowledge of various types of functions and their representation let us move towards some of the solved questions for better practice of the topic.

Example 1: Find the type of one-one function f: {1, 2, 3} → {1, 2, 3}?

1) Into

2) Onto

3) Both onto and into

4) None of the above

Solution: Since f is one – one, three elements of {1, 2, 3} must be taken to 3 different elements of the co – domain {1, 2, 3} under f. Hence, f has to be onto.

Therefore option 2 is correct.

Example 2: The number of many-to-one functions from set A = {6, 7} to set B = {2, 3, 4, 5} is _____.

Solution: Here, n(A) = 2, n(B) = 4

Total no. of functions from set A to B = \(n(B)^{n(A)}=4^{2}=16\).

One to one functions from set A to B = \(^{n(B)}P_{n(A)}=^4P_2=\frac{4!}{2!}=12\).

Number of many-to-one functions = Total no. of functions – One-to-one functions = 16 – 12 = 4.

Example 3: Which of the following is not an “into” function?

1) f (x) = x, ∀ x ∈ A, where f is a function on A = {1, 2, 3, 4, 5}.

2) f (x) = x + 1, ∀ x ∈ A, where f is a function from A = {1, 2, 3, 4, 5} to B = {2, 3, 4, 5, 6, 7}.

3) f (x) = 2x, ∀ x ∈ A, where f is a function from A = {1, 2, 3, 4, 5} to B = {2, 4, 6, 8, 10, 12}.

4) Both 1 and 2

5) None of these

Solution: Concept:

Into Function:

Any function f: A → B is said to be into function if there exists at least one element in B which does not have a pre-image in A, then the function f is said to be an into function.

i.e If Range of function f ⊂ Co-domain of function f, then f is into.

Identity Function:

Any function f: A → A is said to be an identity function if f (x) = x, ∀ x ∈ A.

Note: Any identity function on a non-empty set A, is both one-one and onto.

Onto Function / Surjective Function:

A function f: A → B is said to be an onto function if each element in B has at least one pre-image in A.

i.e If Range of function f = Co-domain of function f, then f is onto

Calculation:

Let us start checking all the options to reach the ans.

Option 1:

Given: f (x) = x, ∀ x ∈ A, where A = {1, 2, 3, 4, 5} and f is a function on A.

 

As we know that the given function is an identity function which is both one-one and onto.

∴ The given function f(x) = x, is not an into function.

Option 2:

Given: f (x) = x + 1, ∀ x ∈ A, where f is a function from A = {1, 2, 3, 4, 5} to B = {2, 3, 4, 5, 6, 7}.

⇒ f(1) = 2 ∈ B, f(2) = 3 ∈ B, f(3) = 4 ∈ B, f(4) = 5 ∈ B, f(5) = 6 ∈ B.

So, Codomain of the given function is B = {2, 3, 4, 5, 6, 7} and range of the function f is {2, 3, 4, 5, 6}

⇒ Here the Range of function f ⊂ Co-domain of function f

Hence, the given function is an “into” function.

Option 3:

Given: f (x) = 2x, ∀ x ∈ A, where f is a function from A = {1, 2, 3, 4, 5} to B = {2, 4, 6, 8, 10, 12}

⇒ f(1) = 2 ∈ B, f(2) = 4 ∈ B, f(3) = 6 ∈ B, f(4) = 8 ∈ B, f(5) = 10 ∈ B.

So, Codomain of the given function is B = {2, 4, 6, 8, 10, 12} and range of the function f is {2, 4, 6, 8, 10}

⇒ Here also the Range of function f ⊂ Co-domain of function f.

Hence, the given function is an “into” function.

Therefore option 1 will be the correct answer.

Example 4: Which of the following options is an identity function?

1) f (x) = 1

2) f (x) = IxI

3) f (x) = 0

4) f (x) = x

Solution: Calculation:

An identity function is a function that always returns the same value that was used as its argument.

Hence f:R→R, f (x) = x is an identity function

Therefore option 4 is the correct answer.

Example 5: Which one of the following is the second-degree polynomial function f(x) where f(0) = 5, f(-1) = 10 and f(1) = 6?

1)\(5x^{2} – 2x + 5\)

2) \(3x^{2} – 2x – 5\)

3) \(3x^{2} – 2x + 5\)

4) \(3x^{2} – 10x + 5\)

5) None of these

Solution: Concept:

General form of a second degree polynomial function is:\(P(x)=a_0+a_1\times x+a_{2}\times x^{2}\), where \(a_{0},a_{1}\text{ and }a_{2}\) are real coefficients and\(a_{2}\ne0\).

Calculation:

As we know, the general form of a second-degree polynomial function is \(P(x)=a_0+a_1\times x+a_{2}\times x^{2}\), where \(a_{0}, a_{1}\text{ and }a_{2}\) are real coefficients and\(a_{2}\ne0\).

Let \(f(x)=a\times x^{2}+b\times x+c\) where a, b and c are real coefficients and a ≠ 0.

Given: f(0) = 5, f(-1) = 10 and f(1) = 6.

⇒ f(0) = c = 5

⇒ f(- 1) = a – b + 5 = 10 —-(1)

⇒ f(1) = a + b + 5 = 6 —(2)

By adding (1) and (2), we get

⇒ 2a + 10 = 16 ⇒ a = 3.

Now by substituting a = 3 in equation (1) we get,

⇒ 3 – b + 5 = 10 ⇒ b = – 2.

Hence,\(f(x)=a\times x^{2}+b\times x+c=3x^{2}-2x+5\) is the correct answer.

Example 6: What will be the value of b, for which the least value of quadratic function \(f(x)=x^{2}-4bx+9\) is -27?

Solution: The given quadratic function is:

\(f(x)=x^{2}-4bx+9\) —- (i)

To obtain the least value of a quadratic function, we need to equate the derivative of f(x) with zero.

So, from equation (i), we get:

\(f^{\prime}(x)=2x-4b=0\)

⇒ x = 2b —-(ii)

Now, substituting the value of f(x) = -27 and x = 2b into equation(i), we get:

\(-27=(2b)^2-(4\times b\times2b)+9\)

\(\Rightarrow-27=-4b^2+9\)

\(\Rightarrow b^2=9\)

\(\Rightarrow b=3,-3\)

∴ The value of b, for which the least value of quadratic function \(f(x)=x^{2}-4bx+9\) is -27; is 3 or -3.

We hope you found this article on Types of Functions informative in consolidating your knowledge and preparing you for U.S. competitive exams like the SAT, ACT, and GRE. Understanding different types of functions is crucial to solving algebra and problem-solving problems efficiently. Keep following us for more information on similar Mathematics topics and visit our practice tests to gauge and enhance your knowledge in preparation for these exams.