Types of Functions

Q: How do I quickly check if a function is one-one?

Use monotonicity: if f prime is always positive or always negative on the domain, the function is strictly monotonic and hence one-one. Or use the horizontal line test on the graph, or solve f(x1)=f(x2) and see if it forces x1=x2.

Q: What is the difference between onto and into functions?

Onto means every element of the co-domain has a pre-image (range equals co-domain). Into means some element of the co-domain has no pre-image (range is a proper subset of co-domain).

Q: Is every polynomial function one-one?

No. Polynomials of even degree on R always have a local extremum, so they are many-one. Polynomials of odd degree may be one-one (like x cubed) or many-one (like x cubed minus 3x).

Q: Can a function be both one-one and many-one?

No. These are mutually exclusive. A function is either one-one (each image has a unique pre-image) or many-one (some image has multiple pre-images).

Q: Why is a bijective function important?

Only bijective functions have inverses. If f from X to Y is bijective, then f-inverse from Y to X exists and is also bijective. Many JEE problems reduce to checking bijectivity before finding the inverse.

Q: Are all even functions many-one?

Yes, whenever the domain contains a non-zero element. An even function satisfies f(-x)=f(x), so x and -x share the same image whenever x is not zero, making the function many-one on any symmetric domain.

Q: How does changing the co-domain affect a function's type?

Restricting the co-domain to match the range converts an into function into an onto function without changing the rule. For example, f(x)=x squared is into as R to R but onto as R to [0, infinity).

MathsFunctionsFor JEE aspirants

Types of functions in JEE Maths classify a function $f : X \to Y$ by how it maps elements. A function is one-one (injective) if distinct inputs give distinct outputs, and many-one otherwise. It is onto (surjective) if every element of the co-domain has a pre-image (range = co-domain), and into if some element has no pre-image. A function that is both one-one and onto is called bijective. This concept covers all these types with tests, examples, and worked problems.

Key Formulas - Quick Reference

One-one (injective): $f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$
Many-one: $\exists x_{1} \neq = x_{2}$ with $f (x_{1}) = f (x_{2})$
Onto (surjective): $\forall y \in Y, \exists x \in X$ with $f (x) = y$ , i.e. range $=$ co-domain
Into: range $⊊$ co-domain (proper subset)
Bijective: one-one AND onto
Horizontal line test: any horizontal line meets the graph at most once $\Leftrightarrow$ one-one

1. One-One (Injective) Function

A function $f : X \to Y$ is called one-one or injective if each element in the domain has a distinct image in the co-domain. Equivalently, if $x_{1} \neq = x_{2}$ then $f (x_{1}) \neq = f (x_{2})$ ; or in the contrapositive form, $f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$ .

Example:

f : R \to R

given by

f (x) = 2 x

is one-one. If

2 x_{1} = 2 x_{2}

then

x_{1} = x_{2}

2. Many-One Function

A function $f : X \to Y$ is called many-one if there exist at least two distinct elements $x_{1} \neq = x_{2}$ in the domain with the same image, that is $f (x_{1}) = f (x_{2})$ .

Example:

f : R \to R

given by

f (x) = x^{2}

is many-one, since

f (2) = f (- 2) = 4

Figure 1: One-one (injective) versus many-one function

Methods to Determine One-One vs Many-One

Algebraic: Assume $f (x_{1}) = f (x_{2})$ . Simplify. If this forces $x_{1} = x_{2}$ , the function is one-one. If it produces another solution, the function is many-one.
Monotonicity: If $f$ is strictly increasing or strictly decreasing on its entire domain, it is one-one.
Horizontal line test: If every horizontal line cuts the graph of $y = f (x)$ at most once, $f$ is one-one; if some horizontal line meets it more than once, $f$ is many-one.
Any continuous function with at least one local maximum or minimum in its domain is many-one.
All even functions (with $x \neq = 0$ in their domain) are many-one since $f (- x) = f (x)$ .
Every polynomial of even degree on $R$ has at least one local extremum, so it is many-one on $R$ . Polynomials of odd degree can be either.

3. Onto (Surjective) Function

A function $f : X \to Y$ is onto or surjective if every element of $Y$ is the image of some element of $X$ . That is, for every $y \in Y$ there exists $x \in X$ with $f (x) = y$ . Equivalently, range $=$ co-domain.

Example:

f : R \to [- 1, 1]

given by

f (x) = sin x

is onto, because every value in

[- 1, 1]

is achieved by some real

x

4. Into Function

A function $f : X \to Y$ is into if at least one element of the co-domain $Y$ is not the image of any element of $X$ . Equivalently, range is a proper subset of co-domain.

Example:

f : R \to R

given by

f (x) = x^{2}

is into, because negative reals in the co-domain are never achieved.

Figure 2: Onto (surjective) versus into function - range vs co-domain

5. Bijective (One-One Onto) Function

A function $f : X \to Y$ is bijective or one-one onto if:

Distinct elements of $X$ have distinct images in $Y$ (one-one), AND
Every element of $Y$ has at least one pre-image in $X$ (onto).

Example:

f : R \to R

given by

f (x) = 2 x

is bijective. It is one-one (strictly increasing) and onto (every real is

2 \cdot (y /2)

Why bijective matters: A function is invertible if and only if it is bijective. This is covered in the next concept.

6. Quick Comparison

Type	Condition	Example
One-one (Injective)	$f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$	$f (x) = 2 x + 1$
Many-one	Some $x_{1} \neq = x_{2}$ share image	$f (x) = x^{2}$
Onto (Surjective)	Range $=$ co-domain	$f : R \to [- 1, 1], f (x) = sin x$
Into	Range $\neq =$ co-domain	$f : R \to R, f (x) = e^{x}$
Bijective	One-one AND onto	$f : R \to R, f (x) = x^{3}$

7. Solved Examples

Solved Example 1

f : R \to R

is defined by

f (x) = \frac{x + 1}{x ^{2} + x + 1}

, is

f

one-one or many-one?

Solution:

Assume $f (x_{1}) = f (x_{2})$ : $\frac{x _{1} + 1}{x _{1}^{2} + x _{1} + 1} = \frac{x _{2} + 1}{x _{2}^{2} + x _{2} + 1}$ .

Cross-multiplying and simplifying leads to $(x_{1} - x_{2}) \cdot g (x_{1}, x_{2}) = 0$ , where $g$ is a polynomial in $x_{1}$ and $x_{2}$ . One solution is $x_{1} = x_{2}$ , but we must check whether $g (x_{1}, x_{2}) = 0$ can give another valid solution.

Test: $f (1) = \frac{2}{3}$ and $f (- 1/2) = \frac{1/2}{1/4 - 1/2 + 1} = \frac{1/2}{3/4} = \frac{2}{3}$ .

So $f (1) = f (- 1/2)$ but $1 \neq = - 1/2$ . Both lie in the domain $R$ .

$f$ is many-one.

Solved Example 2

Let

f : R \to [2, \infty)

be defined by

f (x) = x^{2} + 2 x + b

. If

f

is onto, find the value of

b

Solution:

Complete the square: $f (x) = (x + 1)^{2} + (b - 1)$ .

Since $(x + 1)^{2} \geq 0$ , the range of $f$ is $[b - 1, \infty)$ .

For $f$ to be onto $[2, \infty)$ , we need range $=$ co-domain:

$b - 1 = 2 \Rightarrow b = 3$ .

Solved Example 3

f : R \to R

defined by

f (x) = x^{3}

bijective?

Solution:

One-one: If $x_{1}^{3} = x_{2}^{3}$ then taking cube roots gives $x_{1} = x_{2}$ . Yes.

Onto: For any $y \in R$ , $x = y^{1/3}$ gives $f (x) = y$ . Yes.

$f$ is bijective.

Common Mistakes to Avoid

Watch out

Assuming $f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$ without actually solving. Always test with concrete numbers if the algebra gets messy.
Confusing range and co-domain. A function can only be onto with respect to a specified co-domain. Change the co-domain and the answer changes.
Assuming $f (x) = x^{n}$ is one-one for all $n$ . It is one-one for odd $n$ on $R$ but many-one for even $n$ .
Missing the horizontal line test. If any horizontal line meets the graph twice, the function is many-one, no algebra needed.
Forgetting that continuous functions with a local max or min are automatically many-one.

Frequently Asked Questions

Q1. What are the main types of functions in JEE?

The main classifications are one-one (injective), many-one, onto (surjective), into, and bijective (both one-one and onto). Every function fits somewhere in this scheme depending on how it maps domain elements to co-domain elements.

Q2. How do I quickly check if a function is one-one?

Use monotonicity: if $f^{'} (x) > 0$ or $f^{'} (x) < 0$ throughout the domain, $f$ is strictly monotonic and hence one-one. Alternatively, use the horizontal line test on the graph, or solve $f (x_{1}) = f (x_{2})$ and check if it forces $x_{1} = x_{2}$ .

Q3. What is the difference between onto and into functions?

Onto means every element of the co-domain has at least one pre-image (range $=$ co-domain). Into means at least one element of the co-domain has no pre-image (range is a proper subset of co-domain). Every function is either onto or into with respect to a given co-domain.

Q4. Is every polynomial function one-one?

No. Polynomials of even degree on $R$ always have a local extremum, so they are many-one. Polynomials of odd degree may be one-one (like $x^{3}$ ) or many-one (like $x^{3} - 3 x$ ).

Q5. Can a function be both one-one and many-one?

No. These are mutually exclusive. A function is either one-one (every image has a unique pre-image) or many-one (at least one image has multiple pre-images).

Q6. Why is a bijective function important?

Only bijective functions have inverses. If $f : X \to Y$ is bijective, then $f^{- 1} : Y \to X$ exists and is also bijective. Many JEE problems reduce to checking bijectivity before finding the inverse.

Q7. Are all even functions many-one?

Yes, as long as the domain contains a non-zero element. An even function satisfies $f (- x) = f (x)$ , so $x$ and $- x$ share the same image whenever $x \neq = 0$ , making the function many-one on any domain symmetric about the origin.

Q8. How does changing the co-domain affect a function's type?

Restricting the co-domain to match the range converts an "into" function into an "onto" function without changing the rule. For example, $f (x) = x^{2}$ is into as $R \to R$ but onto as $R \to [0, \infty)$ .

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