How do I rapidly test if a function $f(x)$ is one-one (injective) under JEE timed conditions?

Check the derivative $f'(x)$ across continuous intervals of domain. If $f'(x) > 0$ strictly or $f'(x) < 0$ strictly throughout, the function is strictly monotonic and thus one-one. Graphically, execute the horizontal line test: if every horizontal line intersects $y = f(x)$ at most once, it is injective.

What is the key difference between Codomain and Range in problem solving?

Codomain is the target set $B$ specified in definition $f: A \to B$. Range is the subset of actual outputs $\{f(x) \mid x \in A\}$. A function is onto (surjective) if and only if $\text{Range} = \text{Codomain}$. If $\text{Range} \subset \text{Codomain}$ strictly, $f(x)$ is into.

Why does a composite function $(f \circ g)(x)$ fail to exist even when formula simplifies algebraically?

Composite function $(f \circ g)(x)$ requires output of inner function $g(x)$ to be acceptable input for outer function $f(x)$. If $\text{Range}(g) \cap D_f = \emptyset$, no real values pass through the machine, making $(f \circ g)(x)$ undefined regardless of algebraic formula.

Relations and Functions Notes, Formulas, Traps & PYQs | ExamiFly

Q: What is the key difference between Codomain and Range in problem solving?

Codomain is the target set $B$ specified in definition $f: A \to B$. Range is the subset of actual outputs $\{f(x) \mid x \in A\}$. A function is onto (surjective) if and only if $\text{Range} = \text{Codomain}$. If $\text{Range} \subset \text{Codomain}$ strictly, $f(x)$ is into.

Q: Why does a composite function $(f \circ g)(x)$ fail to exist even when formula simplifies algebraically?

Composite function $(f \circ g)(x)$ requires output of inner function $g(x)$ to be acceptable input for outer function $f(x)$. If $\text{Range}(g) \cap D_f = \emptyset$, no real values pass through the machine, making $(f \circ g)(x)$ undefined regardless of algebraic formula.

Beta

Please wait while we prepare everything for you

Relations and Functions Notes, Formulas, Traps & PYQs | ExamiFly | Examifly

Solved High-Yield Sample Questions

Practice Mode

Question #1JEE Main 2024

Let $A = \{1, 2, 3, 4, 5\}$ and $R$ be a relation on $A$ defined by $R = \{(a, b) \in A \times A : |a^2 - b^2| \text{ is divisible by } 4\}$ . The number of elements in $R$ is:

Solution Locked

Step-by-Step Solution & Concept

Step 1

Analyze parity behavior of squares modulo $4$ . Note $a^2 \equiv 0 \pmod 4$ for even numbers $\{2, 4\}$ and $a^2 \equiv 1 \pmod 4$ for odd numbers $\{1, 3, 5\}$ .

Step 2

Partition set $A$ into odd elements $O = \{1, 3, 5\}$ ( $|O| = 3$ ) and even elements $E = \{2, 4\}$ ( $|E| = 2$ ). Divisibility by $4$ requires both $a, b \in O$ or both $a, b \in E$ .

Step 3

Calculate total pairs in $R$ as $|O \times O| + |E \times E| = 3^2 + 2^2 = 9 + 4 = 13$ elements.

Select an option & click Check Answer to reveal solution

Question #2JEE Main 2023

The domain of definition of the real-valued function $f(x) = \sqrt{\log_{1/2}(x^2 - 3x + 3)}$ is:

Solution Locked

Step-by-Step Solution & Concept

Step 1

For square root stability in $\mathbb{R}$ , non-negativity demands $\log_{1/2}(x^2 - 3x + 3) \ge 0$ , alongside log argument positivity $x^2 - 3x + 3 > 0$ .

Step 2

Reversing inequality for log base $\frac{1}{2} < 1$ gives $x^2 - 3x + 3 \le (1/2)^0 = 1$ .

Step 3

Solve quadratic inequality $x^2 - 3x + 2 \le 0 \implies (x-1)(x-2) \le 0$ , yielding $x \in [1, 2]$ . Since discriminant of $x^2 - 3x + 3 > 0$ is $\Delta = -3 < 0$ , argument is positive $\forall x \in \mathbb{R}$ . Domain is strictly $[1, 2]$ .

Select an option & click Check Answer to reveal solution

If you want to explore more chapters of Mathematics then:

Click Here

Mastering Relations and Functions: Formulas, Traps & Exam Strategy | ExamiFly

Chapter Overview

Core Subtopics & Syllabus Breakdown

Key Formulas & High-Yield Equations

Number of Relations and Functions

Domain of Composite Function

Equivalence Relation Axioms

Condition for Invertibility

Even-Odd Decompositions

Common Mistakes & Traps to Avoid

Past Year Question (PYQ) Trends

Solved High-Yield Sample Questions

Frequently Asked Questions

Master Relations and Functions with 250+ Curated JEE Drills & AI Error-Correction