Complete Function Notes For JEE, NIMCET, CUET And NDA With Definitions, Types, Domain, Range, Formulas, Graphs And Solved Examples. Perfect For Exam Preparation.

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Function – Complete Chapter Notes for JEE Main, N NIMCET, CUET

Table of Contents

1. Definition of Function
2. Representation of Functions
3. Types of Functions
4. Domain, Co-domain & Range
5. Composition of Functions
6. Inverse of a Function
7. Even and Odd Functions
8. Periodic Functions
9. Algebra of Functions
10. Important Standard Functions
11. Graph Transformations
12. Important Examples
13. Summary Points (Exam-Oriented)

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1. Definition of Function

A function is a rule that assigns every element of a non-empty set domain to exactly one element of a non-empty set co-domain.

We write the function as $f : A \to B$.

Example: If $f(x) = x^2$ and $A = \{1,2,3\}$, then range $= \{1,4,9\}$.

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2. Representation of Functions

• Arrow Diagram: mapping arrows from domain to co-domain.
• Rule Method: e.g. $f(x) = 2x + 1$.
• Graph: set of all points $(x, f(x))$.

Vertical Line Test: A vertical line should intersect graph at most once.

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3. Types of Functions

3.1 One-One (Injective)

$f(x_1) = f(x_2) \Rightarrow x_1 = x_2$

3.2 Many-One

Multiple inputs give same output.

3.3 Onto (Surjective)

Range = Co-domain.

3.4 Bijective

One-one and onto → Inverse exists.

3.5 Constant Function

$f(x) = c$

3.6 Identity Function

$I(x) = x$

3.7 Modulus Function

$f(x) = |x|$

3.8 Greatest Integer Function

$f(x) = \lfloor x \rfloor$

3.9 Fractional Part Function

$\{x\} = x - \lfloor x \rfloor$

3.10 Signum Function

$\text{sgn}(x) = 1, 0, -1$ based on sign of $x$

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4. Domain, Co-domain & Range

4.1 Domain

• Denominator $\ne 0$
• Square root → expression $\ge 0$
• Log → argument $> 0$
• Inverse trigonometric → restricted inputs

Example (Domain):

For $f(x)=\frac{1}{\sqrt{5-2x}}$, condition is:

$5-2x > 0 \Rightarrow x < \frac{5}{2}$

Domain: $(-\infty, \frac{5}{2})$

4.2 Range

Actual output values of $f(x)$.

Example (Range):

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

$x^2 \ge 0 \Rightarrow x^2+1 \ge 1$

Range → $(0,1]$

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5. Composition of Functions

If $f : A\to B$ and $g : B\to C$, then $(g\circ f)(x)=g(f(x))$.

Example: If $f(x)=x+2$ and $g(x)=x^2$, then $(g\circ f)(x)=(x+2)^2$.

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6. Inverse of a Function

Inverse exists only for bijective functions.

Steps:

1. Write $y=f(x)$
2. Swap $x,y$
3. Solve for $y$ → this is $f^{-1}(x)$

Example: $f(x)=\frac{2x-1}{3}$

$f^{-1}(x)=\frac{3x+1}{2}$

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7. Even and Odd Functions

Even Function:

$f(-x)=f(x)$

Odd Function:

$f(-x)=-f(x)$

Example: $x^2$ even, $x^3$ odd

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8. Periodic Functions

A function $f$ is periodic if $f(x+T)=f(x)$.

• $\sin x$, $\cos x$ → $2\pi$
• $\tan x$ → $\pi$

Shortcut: For $f(x)=\sin(ax+b)$ → Period $=\frac{2\pi}{|a|}$

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9. Algebra of Functions

• $(f+g)(x)=f(x)+g(x)$
• $(f-g)(x)=f(x)-g(x)$
• $(fg)(x)=f(x)g(x)$
• $(kf)(x)=k\cdot f(x)$
• $\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}$, $g(x)\ne 0$

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10. Important Standard Functions

• Polynomial: $f(x)=ax^n+\cdots$
• Rational: $f(x)=\frac{P(x)}{Q(x)}$
• Exponential: $a^x$
• Logarithmic: $\log_a x$
• Trigonometric, inverse trigonometric
• Modulus, step, signum

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11. Graph Transformations

• Vertical shift: $f(x)+k$
• Horizontal shift: $f(x-a)$
• Reflection in x-axis: $-f(x)$
• Reflection in y-axis: $f(-x)$
• Vertical stretch: $k f(x)$
• Horizontal stretch: $f(kx)$

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12. Important Examples

Example 1: One-One / Many-One

$f(x)=x^2$ is many-one on $\mathbb{R}$ but one-one on $[0,\infty)$.

Example 2: Range

$f(x)=\sqrt{9-x^2}$ → Range $=[0,3]$

Example 3: Composition

For $f(x)=x^2+1$, $g(x)=\sqrt{x-1}$:

$(g\circ f)(x)=|x|$

Example 4: Inverse

$f(x)=\frac{2x-1}{3}$ → $f^{-1}(x)=\frac{3x+1}{2}$

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13. Summary Points

• If $f$ is increasing → it is injective.
• Only bijective → inverse exists.
• For log → argument $>0$.
• For mod function → use piecewise.
• For range → convert to standard forms.
• Graphs of G.I.F., modulus, signum → must memorize.

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FAQs About NIMCET Notes, CUET Notes & Best Test Series

Q1. Are these NIMCET notes suitable for beginners?

Yes, the NIMCET notes are written in a step-by-step manner starting from basics to advanced concepts, making them perfect for beginners as well as repeaters.

Q2. Do these notes cover the complete NIMCET syllabus?

Yes, all important chapters of Mathematics, Logical Reasoning, and Computer Awareness are covered as per the latest NIMCET syllabus.

Q3. Are CUET MCA notes different from NIMCET notes?

Mostly the concepts are similar, but CUET MCA also includes extra reasoning and aptitude topics. Separate CUET-focused notes are provided for those parts.

Q4. Do I need separate notes for CUET and NIMCET?

Basic concepts are common, but exam patterns differ. So using common notes for concepts along with separate CUET-specific practice is recommended.

Q5. Are these notes enough to crack NIMCET?

Yes, if you follow the notes, solve PYQs, and give regular mock tests, you can easily score 650+ and get a top NIT.

Q6. Are PYQs included in the notes?

Yes, past year questions of NIMCET and CUET MCA are included with solutions for better practice.

Q7. How many times should I revise these notes?

Revision at least 3–4 times is recommended before the exam for maximum retention.

Q8. Are these notes available in PDF format?

Yes, NIMCET and CUET notes are available in clean, printable PDF format chapter-wise.

Q9. Which is the best test series for NIMCET?

The best test series is the one that provides:

• NIMCET-level questions
• Sectional tests
• Full-length mocks
• Detailed solutions
• AIR-based ranking

Aspire Study Test Series is highly recommended for NIMCET-level difficulty.

Q10. How many mock tests should I attempt before NIMCET?

Around 20–30 full-length tests and 50+ chapter tests are ideal.

Q11. Is CUET MCA test series different from NIMCET test series?

Yes. CUET test series has more questions from Reasoning, DI, and Aptitude compared to NIMCET’s math-heavy pattern.

Q12. Which test series is best for CUET MCA?

A CUET MCA-oriented test series that includes reasoning, DI, quantitative aptitude, and CS topics with full-length papers is recommended.

Q13. Do test series help in time management?

Absolutely. Test series improve accuracy, speed, question selection ability, and exam temperament.

Q14. Are solutions provided for all mock tests?

Yes, every mock test includes detailed step-by-step explanations for easy understanding.

Q15. Can I attempt mock tests on mobile?

Yes, all tests can be attempted on mobile, tablet, and laptop with auto-save features.