Types of Functions in Math: Definition, Examples, Composition & Exam Tips

Types of functions in Math are classified based on various criteria including mapping properties, equation forms, and range characteristics. Key types include one-to-one, many-to-one, onto, and into functions based on element mapping. Additionally, functions are categorized based on equation form, such as linear, quadratic, cubic, and polynomial functions. Other important types include rational, modulus, and even/odd functions based on their range and behaviour. NCERT class 11 notes on relations and functions cover the different types of functions in detail.These classifications help in understanding the properties and applications of different types of functions in mathematical problems. 

One of the first types of functions if Injective function. Basically, a function f: A $\to$ B is one - one function or injective function when distinct elements of A have distinct images in B.

Thus, f: A $\to$ B is one - one function $\iff$ f(a) = f(b) $\Rightarrow$ a = b $\forall$ a, b $\in$ A

$\iff$  a $\ne$ b $\Rightarrow$  f(a) $\ne$  f(b) $\forall$ a, b $\in$ A.

NCERT excercise of relations and functions include questions based this function type. Student must practice the excercise before sitting in the CBSE board exams.

Here,  function f(x) = x+1 is a type of one-to-one function as it produces a different answer for every input.

The function f(x) = x², on the other hand, is not a one-to-one function as it gives the same answer for more than one input.

f (2) = 4 and f (–2) = 4

A function f: A $\to$ B is onto function or a surjective function if every element of B is the f - image of some element of A.

$\Rightarrow$ f(A) = B or range of f is the co – domain of  f.

Thus, f: A $\to$ B is onto function $\iff$ f(A) = B i.e., range of f = co – domain of f.

Let us consider an example to understand this function type:

f(x) = x+1:  If the domain and co - domain are both the set of real numbers (R), this function is onto.

f(x) = 2x: If the domain and co - domain are both the set of real numbers (R), this function is onto.

f(x) = x mod 3:  This function maps integers to 0,1,2 and is onto as every element in 0,1,2 has at least one corresponding integer in the domain.

f(x) = x²:  This function maps real numbers to non - negative real numbers (R⁺) is onto if the co-domain is restricted to the non-negative real numbers (R⁺).

Also known as Bijective function or Bijection; a function f: A $\to$ B is said to be one – one onto function or bijective function if it is both one - one and onto i.e., if the distinct elements of A have distinct images in B and each element of B is the image of some elements of A. Let us consider an example to understand one-one onto function type:

Define $f:\mathbb{R}\to \mathbb{R}$ by $f\left(x\right)=2x+1$.

1. One-to-One (Injective): $$f(x₁)=f(x₂)⟹2x₁+1=2x₂+1⟹x₁=x₂$$
2. Onto (Surjective):

   For any $y$ in $\mathbb{R}$, solve $y=2x+1$ for $x=\frac{y-1}{2}$.

   Since $\frac{y-1}{2}$ belongs to $\mathbb{R}$, every $y$ has a pre-image.

Therefore, $f\left(x\right)=2x+1$ is bijective.

Important reads for students:

Another type of function in Math is Bijective function. A function f: A $\to$ B is said to be one - one and onto function (or bijective function) if it is both one - one and onto. Those who plan to sit in entrance examinations like NEET and CUET must learn about these functions in detail so that they can answer problems based on this topic. 

Let us consider the function f: R $\to$ R, defined as f(x) = 2x + 1, where R represents the set of all real numbers.

Injective (one - one) function: If f(x₁) = f(x₂),

Then 2 x₁ + 1 = 2 x₂ + 1.

Solving for x₁, we get x₁ = x₂, meaning distinct inputs lead to distinct outputs.

Surjective (Onto) function: For any y in R, we can find an x in R such that f(x) = y.

Solving for x, we get x = (y –2) / 2, which is a real number for any real number y.

Since the function is both injective and surjective, it is a bijection.

The last type of function is invertible function. As per the definition, a function f: A $\to$ B is said to invertible if there exists a function g: B $\to$ A such that gof = I_A and fog = I_B where I_A and I_B are identity functions. The function g is called the inverse of f and is denoted by f^-1.

Entrance exams like IIT JAM and JEE Main cover questions from this function type which makes it important for students to practice them. To understand this function, below given example is a good fit.

• f(x) = 2x + 3: This linear function is invertible because for every x, there’s a unique y, and for every y, there’s unique x.

The inverse function is f^-1(x) = (x–3)/2.

• f(x) = x³: This cubic function is invertible because it’s both one - to - one and onto. Its inverse is f^-1(x) = ³√x.
• f(x) = (3x – 4)/5: This rational function is invertible. Its inverse is f^-1(x) = (5x + 4)/3.
• f(x) = ax + b: This general linear function is invertible, where a $\ne$ 0, and its inverse is f^-1(x) = (x – b)/a.

Composition of two functions means that the output of first function type becomes the input of the second type of function. Let f: A $\to$ B and g: B $\to$ C be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof: A $\to$ C given by (gof)(x) = g(f(x)) $\forall$ x $\in$ A.

Examples:

f(x) = 2x + 3

g(x) = x²

f(g(x)) = f(x²) = 2(x²) + 3 = 2 x² + 3

g(f(x)) = g(2x + 3) = (2x + 3)² = 4 x² + 12x + 9

h(x) = sinx

k(x) = x – $\pi$

h (k(x)) = h (x – $\pi$ ) = sin (x – $\pi$ )