1. Algebra — Relations and Functions

One-one functions

Q: How can I quickly distinguish between one-one and many-one functions in exam questions?

Check the derivative $f'(x)$. If $f'(x)$ is strictly greater than zero or strictly less than zero across the entire domain, the function is one-one. If $f'(x)$ changes sign or equals zero over an interval, it is many-one.

Q: Why do polynomial functions of even degree fail to be one-one on $\mathbb{R}$?

By Rolle's Theorem or fundamental properties of continuous functions, even-degree polynomials on $\mathbb{R}$ must possess at least one turning point where the derivative is zero and changes sign, violating strict monotonicity.

Q: What is the best shortcut for fraction/rational function injectivity?

Compute the cross-product $f(x_1) = f(x_2)$ or use the numerator determinant condition $ad - bc \neq 0$ for linear fractional functions.

A function $f: A \to B$ is one-one (injective) if distinct elements in the domain map to distinct images in the codomain, meaning $f(x_1) = f(x_2) \implies x_1 = x_2$.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Injectivity (One-One) Definition & Monotonicity Criterion

Let $f: A \to B$ be a function. Then $f$ is one-one (injective) if and only if for all $x_1, x_2 \in A$, $f(x_1) = f(x_2) \implies x_1 = x_2$, or equivalently, $x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$. Furthermore, if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ with $f'(x) > 0$ \text{ (or } f'(x) < 0) \forall x \in (a, b)$, then $f$ is strictly monotonic and hence strictly one-one on $[a, b]$.

📐 Geometric Interpretation

Every horizontal line $y = k$ intersects the graph of a one-one function at most once. Visually, the curve is strictly increasing or strictly decreasing, never turning back on itself.

⚙️ Algebraic & Structural Meaning

Injectivity guarantees the existence of a left inverse and enables function inversion over its range, forming the foundational bedrock for inverse trigonometric and logarithmic structures.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Strict Monotonicity on Domain	`Domain interval $A \subseteq \mathbb{R}$`	Strictly Mandatory for Derivative Test	If derivative changes sign (e.g., has local extrema), the function is many-one.
Continuity across Domain	`Connected interval $[a, b]$`	Mandatory for Darboux / Monotonicity link	Discontinuous functions can be injective without being monotonic, but monotonic checks fail.
Domain Restriction Preservation	`Subsets $A_1, A_2 \subset Domain$`	Essential for composite and restricted domain injectivity	Overlapping images destroy injectivity when domains are combined.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Strict Derivative Sign Test

$$\text{If } f'(x) > 0 \text{ or } f'(x) < 0 \text{ on } (a, b) \implies f \text{ is one-one}$$

Rule: Compute $f'(x)$. If the expression can never change sign across the domain, the function is unconditionally one-one.

Application: Directly used in UP PGT to check injectivity of cubic and transcendental polynomials.

Polynomial Degree & Parity Rule

$$f(x) = \text{Polynomial of odd degree with } f'(x) \text{ sign-definite}$$

Rule: Odd-degree polynomials with non-zero derivative discriminants ($\Delta < 0$ for quadratic derivatives) are always one-one from $\mathbb{R} \to \mathbb{R}$.

Application: Frequent in TGT/PGT screening tests for quick options elimination.

Linear Function Slope Test

$$f(x) = ax + b \implies f \text{ is one-one } \iff a \neq 0$$

Rule: For linear functions over $\mathbb{R}$, non-zero slope guarantees injectivity instantly.

Exponential & Logarithmic Monotonicity

$$f(x) = a^x (a>0, a \neq 1) \text{ or } \ln x \text{ are strictly one-one}$$

Rule: Standard transcendental building blocks are inherently one-one on their natural domains.

Application: Used in composite function injectivity analysis.

Sum of One-One Functions

$$f, g \text{ strictly increasing } \implies f+g \text{ is strictly one-one}$$

Rule: The sum or composition of strictly increasing (or decreasing) functions is strictly one-one.

Application: Tested in DSSSB PGT multi-statement questions.

Horizontal Line Test Elimination

$$y = x^2, \sin x, |x| \text{ fail horizontal line test}$$

Rule: Instantly discard functions with even powers, periodic behavior, or absolute values when global injectivity on $\mathbb{R}$ is requested.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Assuming Even Degree Polynomials Can Be One-One on $\mathbb{R}$

Common Pitfall: Candidates apply algebraic checks without realizing even-degree polynomials must have turning points where $f'(x) = 0$ and change sign.

Counterexample: $f(x) = x^2, \quad f(-1) = f(1) = 1$

Why it fails: Fails the horizontal line test and injectivity definition because distinct inputs yield identical outputs.

🛡️ Exam Defense: Never mark an even-degree polynomial as one-one when the domain is the entire set of real numbers $\mathbb{R}$.

⚠️ COMMON ERROR #02

Ignoring Domain Restrictions

Common Pitfall: Judging injectivity solely by the algebraic formula while ignoring the given domain and codomain sets.

Counterexample: $f: \mathbb{R} \to \mathbb{R}, \; f(x) = x^2$

Why it fails: While $f(x) = x^2$ is many-one on $\mathbb{R}$, it is strictly one-one on $[0, \infty)$.

🛡️ Exam Defense: Always examine the domain $A$ and codomain $B$ specification before concluding injectivity.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
One-One (Injective)	$f(x_1) = f(x_2) \implies x_1 = x_2$	$f'(x)$ keeps constant sign	Horizontal line intersects at most once	Words: 'Distinct elements map to distinct images'
Many-One	$\exists x_1 \neq x_2$ with $f(x_1) = f(x_2)$	$f'(x) = 0$ and changes sign	Horizontal line intersects twice or more	Even powers, periodic functions
Onto (Surjective)	Range = Codomain	$f(A) = B$	Graph covers entire codomain strip vertically	Codomain equals range analysis
Bijective	Both One-One and Onto	Invertible mapping $f^{-1}$ exists	1-to-1 pairing between sets	Inverse function existence questions

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Determine whether the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^3 + 5x + 1$ is one-one, many-one, or neither.

(A) One-one(B) Many-one(C) Neither one-one nor onto(D) Constant function

📚 Method A: Formal Textbook Derivation

Compute derivative: $f'(x) = 3x^2 + 5$. Since $3x^2 \ge 0$, $f'(x) \ge 5 > 0$ for all $x \in \mathbb{R}$. The derivative is strictly positive everywhere, hence $f$ is strictly increasing and therefore one-one.

⚡ Method B: 25-Second Exam Shortcut

Odd degree polynomial with strictly positive derivative is instantly classified as one-one.

💡 Tip: Check $f'(x)$ sign; if discriminant of quadratic derivative is negative, derivative never changes sign.

CANONICAL PROBLEM 02 Dual-Method Breakdown

If $f: [0, \infty) \to [0, \infty)$ is given by $f(x) = \frac{x}{1+x}$, then $f$ is:

(A) One-one but not onto(B) Onto but not one-one(C) Both one-one and onto(D) Neither one-one nor onto

📚 Method A: Formal Textbook Derivation

Let $f(x_1) = f(x_2) \implies \frac{x_1}{1+x_1} = \frac{x_2}{1+x_2} \implies x_1 + x_1 x_2 = x_2 + x_1 x_2 \implies x_1 = x_2$. Thus $f$ is one-one. Range is $[0, 1)$, whereas codomain is $[0, \infty)$, so it is not onto.

⚡ Method B: 25-Second Exam Shortcut

Differentiate $f'(x) = \frac{1}{(1+x)^2} > 0$ (one-one). As $x \to \infty$, $f(x) \to 1$ (range bounded, not onto $\mathbb{R}^+$).

💡 Tip: Always check range vs codomain for onto, and derivative sign for one-one.

Practice questions

Choose an answer to check it and see the solution:

Which of the following functions $f: \mathbb{R} \to \mathbb{R}$ is one-one?

A $f(x) = |x|$

B $f(x) = x^2 + 2x + 1$

C $f(x) = x^3 - 3x$

D $f(x) = 2x^3 + 5$

Derivation & Analysis:

$f(x) = 2x^3 + 5$ has derivative $f'(x) = 6x^2 \ge 0$. It is strictly monotonic and one-one. Others fail injectivity.

Let $f: [0, \pi/2] \to \mathbb{R}$ be defined by $f(x) = \sin x$. Then $f$ is:

A One-one

B Many-one

C Constant

D Not continuous

Derivation & Analysis:

On $[0, \pi/2]$, $\cos x > 0$ (except at endpoint), so $\sin x$ is strictly increasing, making it one-one.

If $f: A \to B$ is a one-one function and $n(A) = 4, n(B) = 6$, then the total number of such one-one functions is:

A 24

B 360

C 4096

D 120

Derivation & Analysis:

Number of one-one functions from set of size $m$ to set of size $n$ ($m \le n$) is $P(n, m) = P(6, 4) = \frac{6!}{2!} = 360$.

The function $f(x) = x e^x$ for $x \in \mathbb{R}$ is:

A Many-one

B One-one

C Periodic

D Bounded

Derivation & Analysis:

$f'(x) = e^x + x e^x = e^x(1+x)$. While $f'(x)$ changes sign at $x=-1$, check values: $x e^x = y$. For $x \ge 0$, it is strictly increasing. Across $\mathbb{R}$, analyze $f(x_1) = f(x_2)$. Since $f'(x) > 0$ for $x > -1$ and one can verify monotonicity on restricted intervals, standard calculus shows $f$ is one-one on $\mathbb{R}$.

If $f(x) = \frac{ax+b}{cx+d}$ is one-one on its domain, what condition must hold on constants $a, b, c, d$?

A $ad - bc = 0$

B $ad - bc \neq 0$

C $a + d = 0$

D $b + c = 0$

Derivation & Analysis:

For a linear fractional transformation, $f'(x) = \frac{ad - bc}{(cx+d)^2}$. For injectivity, derivative must never be zero, which requires $ad - bc \neq 0$.

Which function is NOT one-one on $\mathbb{R}$?

A $f(x) = 3x - 4$

B $f(x) = x^3$

C $f(x) = 2^x$

D $f(x) = x^2 - 4x + 3$

Derivation & Analysis:

$f(x) = x^2 - 4x + 3$ is a quadratic (even-degree polynomial equivalent), and $f(1) = f(3) = 0$, so it is many-one.

Let $f: \mathbb{N} \to \mathbb{N}$ be defined by $f(n) = n^2 + n + 1$. Then $f$ is:

A One-one

B Many-one

C Constant

D Not a function

Derivation & Analysis:

For $n_1, n_2 \in \mathbb{N}$, $n_1^2 + n_1 + 1 = n_2^2 + n_2 + 1 \implies (n_1 - n_2)(n_1 + n_2 + 1) = 0$. Since $n_1, n_2 \ge 1$, $n_1 + n_2 + 1 \neq 0$, so $n_1 = n_2$. Thus $f$ is one-one.

If $f(x) = \cos x$ for $x \in [0, \pi]$, then $f$ is:

A One-one

B Many-one

C Not continuous

D Surjective to $\mathbb{R}$

Derivation & Analysis:

On $[0, \pi]$, $\cos x$ is strictly decreasing ($f'(x) = -\sin x < 0$), hence strictly one-one.

The number of one-one functions from a set $A$ containing 3 elements to a set $B$ containing 3 elements is:

A 3

B 6

C 9

D 27

Derivation & Analysis:

$n(A) = 3, n(B) = 3$. Number of one-one functions is $3! = 6$.

Q10

If $f(x) = x|x|$ for $x \in \mathbb{R}$, then $f$ is:

A One-one

B Many-one

C Discontinuous

D Not differentiable at all points

Derivation & Analysis:

$f(x) = x^2$ for $x \ge 0$ and $-x^2$ for $x < 0$. This is strictly increasing across all real numbers ($\sqrt{y}$ is unique), hence it is one-one.

Common questions

Check the derivative $f'(x)$. If $f'(x)$ is strictly greater than zero or strictly less than zero across the entire domain, the function is one-one. If $f'(x)$ changes sign or equals zero over an interval, it is many-one.

By Rolle's Theorem or fundamental properties of continuous functions, even-degree polynomials on $\mathbb{R}$ must possess at least one turning point where the derivative is zero and changes sign, violating strict monotonicity.

Compute the cross-product $f(x_1) = f(x_2)$ or use the numerator determinant condition $ad - bc \neq 0$ for linear fractional functions.