It equals 0 if x is an integer and -1 otherwise. For example [3]+[-3]=0, but [2.4]+[-2.4]=2-3=-1. It is a common JEE identity.

Some Important Properties Of Functions

Q: What is the greatest integer function?

The greatest integer function, written [x] or floor of x, gives the largest integer less than or equal to x. For example [3.7]=3 and [-1.2]=-2. Its domain is R and range is the set of integers.

Q: What is the fractional part of a number?

The fractional part is {x}=x-[x]. It always lies in [0, 1), is periodic with period 1, and equals 0 when x is an integer. Example: {5.68}=0.68; {-2.3}=0.7.

Q: How do I check if a function is even or odd?

Replace x by -x and simplify. If f(-x)=f(x), the function is even. If f(-x)=-f(x), it is odd. Otherwise, it is neither. The domain must be symmetric about 0 for this classification.

Q: Can a function be both even and odd?

Only one function has this property: the zero function f(x)=0. It satisfies both f(-x)=f(x) and f(-x)=-f(x) trivially, and no other function does.

Q: What is the period of sin(ax+b)?

If f(x) has period T, then f(ax+b) has period T divided by absolute value of a. Since sin x has period 2 pi, sin(ax+b) has period 2 pi over absolute value of a. For example, sin(3x) has period 2 pi over 3.

Q: How do I find the period of a sum of periodic functions?

Compute the period of each function separately, then take the LCM of the two periods. The result is the period of the sum, provided no smaller positive T works. If the ratio of the periods is irrational, the sum is not periodic.

Q: Is every constant function periodic?

Yes, a constant function f(x)=c satisfies f(x+T)=c for every positive T, so it is periodic. However, it has no fundamental period because there is no smallest positive T that works.

Q: Why is f(0)=0 for an odd function?

If f is odd and 0 is in the domain, then f(-0)=-f(0), that is f(0)=-f(0), giving 2 f(0)=0 so f(0)=0. If f(0) is not zero, the function cannot be odd.

MathsFunctionsFor JEE aspirants

This concept covers four important properties of functions used across JEE Maths: the greatest integer function $[x]$ (the floor of $x$ ), the fractional part ${x} = x - [x]$ , even and odd functions (defined by $f (- x) = f (x)$ or $f (- x) = - f (x)$ ), and periodic functions (satisfying $f (x + T) = f (x)$ ). Together these tools describe symmetry, repetition, and integer structure of functions - all frequent themes in JEE problems.

Key Formulas - Quick Reference

Greatest integer: $[x] = n$ where $n \leq x < n + 1$ , $n \in Z$
Fractional part: ${x} = x - [x]$ , with $0 \leq {x} < 1$
$[x] + [- x] = 0$ if $x \in Z$ , else $- 1$
${x} + {- x} = 0$ if $x \in Z$ , else $1$
Even function: $f (- x) = f (x)$ ; graph symmetric about $y$ -axis
Odd function: $f (- x) = - f (x)$ ; graph symmetric about origin; $f (0) = 0$ if $0 \in$ domain
Decomposition: every $f$ on a symmetric domain splits as $f (x) = E (x) + O (x)$ , where the even part is $E (x) = \frac{f ( x ) + f ( - x )}{2}$ and the odd part is $O (x) = \frac{f ( x ) - f ( - x )}{2}$
Period of $f (a x + b)$ is $\frac{T}{∣ a ∣}$ if $f$ has period $T$
Period of $sin^{n} x, cos^{n} x$ : $2 π$ if $n$ odd, $π$ if $n$ even; period of $tan^{n} x, cot^{n} x$ : always $π$

1. Greatest Integer Function

Every real number $x$ lies between two consecutive integers: $n \leq x < n + 1$ for some integer $n$ . This unique integer $n$ is called the greatest integer of $x$ and is written $[x]$ (also called the floor).

[x] =

the largest integer that is less than or equal to

x

Examples: $[3.54] = 3$ (since $3 \leq 3.54 < 4$ ). $[- 2.95] = - 3$ (since $- 3 \leq - 2.95 < - 2$ ). $[5] = 5$ (if $x$ is an integer, $[x] = x$ ).

Domain:

R

. Range:

Z

. Non-periodic. Neither even nor odd.

Figure 1: Greatest integer function y = [x] - a step function with domain R and range Z

Key Properties of $[x]$

$[x] = n \Leftrightarrow n \leq x < n + 1$ .
$P_{1} \leq [x] \leq P_{2} \Rightarrow P_{1} \leq x < P_{2} + 1$ , i.e. $x \in [P_{1}, P_{2} + 1)$ .
$x - 1 < [x] \leq x$ .
$[[x]] = [x]$ (idempotent).
$[n + x] = n + [x]$ for any integer $n$ .
$[x] + [- x] = {0 if x \in Z - 1 if x \in / Z$

2. Fractional Part Function

The fractional part of a real number $x$ is defined as ${x} = x - [x]$ .

Examples: If $x = 5.68$ , then $[x] = 5$ and ${x} = 0.68$ . If $x = - 2.3$ , then $[x] = - 3$ and ${x} = 0.7$ .

Domain:

R

. Range:

[0, 1)

. Period:

1

. Neither even nor odd.

Figure 2: Fractional part function y = {x} - periodic sawtooth with period 1 and range [0, 1)

Key Properties of ${x}$

$0 \leq {x} < 1$ for every real $x$ .
If $x$ is an integer, ${x} = 0$ .
$[{x}] = 0$ and ${[x]} = 0$ .
${x} + {- x} = {0 if x \in Z 1 if x \in / Z$
${x}$ is periodic with period $1$ : ${x + 1} = {x}$ .

Solved Example 1

Solve for

x

[x] + \frac{1}{x - 2.7} = 2 + \frac{1}{x - 2.7}

Solution:

Cancelling $\frac{1}{x - 2.7}$ from both sides gives $[x] = 2$ , which means $x \in [2, 3)$ .

But the cancellation is only valid when $\frac{1}{x - 2.7}$ is defined, i.e. $x \neq = 2.7$ .

So the solution set is $x \in [2, 3) - {2.7} = [2, 2.7) \cup (2.7, 3)$ .

3. Even and Odd Functions

Let $f$ be a real-valued function whose domain $D$ satisfies: $x \in D \Rightarrow - x \in D$ (a symmetric domain).

Even function:

f (- x) = f (x)

for every

x \in D

.
Odd function:

f (- x) = - f (x)

for every

x \in D

Figure 3: Even function - graph is symmetric about the y-axis, f(-x) = f(x)

Figure 4: Odd function - graph is symmetric about the origin, f(-x) = -f(x)

Key Properties

Even functions are symmetric about the $y$ -axis: if $(x, y)$ is on the graph, so is $(- x, y)$ .
Odd functions are symmetric about the origin: if $(x, y)$ is on the graph, so is $(- x, - y)$ .
$f (x) = 0$ is the only function that is both even and odd.
If $0 \in$ domain of an odd function, then $f (0) = 0$ . So $f (0) \neq = 0$ implies $f$ is not odd.
Any function $f$ on a symmetric domain can be written as a sum of an even part and an odd part: $f (x) = E (x) + O (x)$ , where $E (x) = \frac{f ( x ) + f ( - x )}{2}$ is the even part and $O (x) = \frac{f ( x ) - f ( - x )}{2}$ is the odd part.
Even $\times$ Even $=$ Even. Odd $\times$ Odd $=$ Even. Even $\times$ Odd $=$ Odd.

Solved Example 2

Classify each as even, odd, or neither: (i)

f (x) = \frac{∣ x ^{2} - 1∣ cos x}{1 + lo g ( e ^{x} + e ^{- x} )}

, (ii)

f (x) = lo g \frac{1 - x}{1 + x}

, (iii)

f (x) = lo g (x + 1 + x^{2})

, (iv)

f (x) = sin x - cos x

Solution:

(i) $f (- x) = \frac{∣ ( - x ) ^{2} - 1∣ cos ( - x )}{1 + lo g ( e ^{- x} + e ^{x} )} = \frac{∣ x ^{2} - 1∣ cos x}{1 + lo g ( e ^{x} + e ^{- x} )} = f (x)$ . So $f$ is even.

(ii) $f (- x) = lo g \frac{1 + x}{1 - x} = - lo g \frac{1 - x}{1 + x} = - f (x)$ . So $f$ is odd.

(iii) $f (- x) = lo g (- x + 1 + x^{2}) = lo g \frac{( - x + 1 + x ^{2} ) ( x + 1 + x ^{2} )}{x + 1 + x ^{2}} = lo g \frac{1}{x + 1 + x ^{2}} = - f (x)$ . So $f$ is odd.

(iv) $f (- x) = - sin x - cos x$ , which is neither $f (x)$ nor $- f (x)$ . So $f$ is neither.

Solved Example 3

f (x + y) = f (x) + f (y)

for all

x, y \in R

, classify

f

Solution:

Put $x = y = 0$ : $f (0) = 2 f (0) \Rightarrow f (0) = 0$ .

Put $y = - x$ : $f (x) + f (- x) = f (0) = 0 \Rightarrow f (- x) = - f (x)$ .

Hence $f$ is odd.

4. Periodic Functions

A function $f : X \to Y$ is called periodic if there exists a positive real number $T$ such that $f (x + T) = f (x)$ for every $x \in X$ . The least such positive $T$ is the fundamental period (or simply period) of $f$ .

Figure 5: Periodic function y = sin x - repeats every 2π

How to Test for Periodicity

Set $f (x + T) = f (x)$ and solve for positive values of $T$ that are independent of $x$ . If such $T$ exist, $f$ is periodic and the smallest one is the period. If no such $T$ exists, $f$ is non-periodic.

Key Properties

A constant function is periodic but has no well-defined fundamental period.
If $f (x)$ has period $T$ , then $f (a x + b)$ has period $\frac{T}{∣ a ∣}$ (for $a \neq = 0$ ).
If $f (x)$ has period $T$ , then $a f (x) + b$ has the same period $T$ (for $a \neq = 0$ ).
If $f$ has period $p = m / n$ and $g$ has period $q = r / s$ (in lowest terms), then $f + g$ (or any algebraic combination) has period equal to the LCM of $p$ and $q$ , provided no smaller positive $T$ works.
LCM of two periods exists only if their ratio is rational. If $p / q$ is irrational, $f + g$ is non-periodic.
$sin^{n} x, cos^{n} x, csc^{n} x, sec^{n} x$ have period $2 π$ if $n$ is odd, and $π$ if $n$ is even.
$tan^{n} x$ and $cot^{n} x$ have period $π$ for every positive integer $n$ .
If $g$ is periodic, then $f \circ g$ is periodic. (The period of $f \circ g$ may or may not equal the period of $g$ .)
If $f$ is periodic and $g$ is strictly monotonic (not linear), then $f \circ g$ is not periodic.

Solved Example 4

Find the period of

f (x) = sin 4 x + tan 2 x

Solution:

Period of $sin 4 x$ is $\frac{2 π}{4} = \frac{π}{2}$ .

Period of $tan 2 x$ is $\frac{π}{2}$ .

Both have the same period $\frac{π}{2}$ , so the period of $f (x)$ is $\frac{π}{2}$ .

Common Mistakes to Avoid

Watch out

Cancelling terms without checking whether they are defined. If $\frac{1}{x - a}$ appears on both sides, $x = a$ must be excluded from the solution set.
Assuming $[- x] = - [x]$ . This is only true when $x$ is an integer. Otherwise $[x] + [- x] = - 1$ .
Assuming ${- x} = - {x}$ . The fractional part is always in $[0, 1)$ , so it can never be negative.
Concluding a function is odd just because $f (0) = 0$ . The condition $f (0) = 0$ is necessary but not sufficient.
Adding periods without taking LCM. Period of $f + g$ is $LCM (T_{f}, T_{g})$ , not $T_{f} + T_{g}$ .
Assuming LCM always exists. If the ratio $T_{f} / T_{g}$ is irrational, $f + g$ has no period.
Ignoring that "less than 1" in $0 \leq {x} < 1$ is strict. The fractional part never equals $1$ .

Frequently Asked Questions

Q1. What is the greatest integer function?

The greatest integer function, written $[x]$ or $⌊ x ⌋$ , gives the largest integer less than or equal to $x$ . For example $[3.7] = 3$ and $[- 1.2] = - 2$ . Its domain is $R$ and range is $Z$ .

Q2. What is the fractional part of a number?

The fractional part is ${x} = x - [x]$ . It always lies in $[0, 1)$ , is periodic with period $1$ , and equals $0$ when $x$ is an integer. Example: ${5.68} = 0.68$ ; ${- 2.3} = 0.7$ .

Q3. How do I check if a function is even or odd?

Replace $x$ by $- x$ and simplify. If $f (- x) = f (x)$ , the function is even. If $f (- x) = - f (x)$ , it is odd. If neither, it is neither. The domain must be symmetric about $0$ for this classification.

Q4. Can a function be both even and odd?

Only one function has this property: the zero function $f (x) = 0$ . It satisfies both $f (- x) = f (x)$ and $f (- x) = - f (x)$ trivially, and no other function does.

Q5. What is the period of $sin (a x + b)$ ?

If $f (x)$ has period $T$ , then $f (a x + b)$ has period $T /∣ a ∣$ . Since $sin x$ has period $2 π$ , $sin (a x + b)$ has period $2 π /∣ a ∣$ . For example, $sin (3 x)$ has period $2 π /3$ .

Q6. How do I find the period of a sum of periodic functions?

Compute the period of each function separately, then take the LCM of the two periods. The result is the period of the sum, provided no smaller positive $T$ satisfies $f (x + T) + g (x + T) = f (x) + g (x)$ . If the ratio of the two periods is irrational, the sum is not periodic.

Q7. Is every constant function periodic?

Yes, a constant function $f (x) = c$ satisfies $f (x + T) = f (x) = c$ for every positive $T$ , so it is periodic. However, it has no fundamental period because there is no smallest positive $T$ that works.

Q8. Why is $f (0) = 0$ for an odd function?

If $f$ is odd and $0 \in$ domain, then $f (- 0) = - f (0)$ , i.e. $f (0) = - f (0)$ , giving $2 f (0) = 0$ so $f (0) = 0$ . This is a necessary check: if $f (0) \neq = 0$ , the function cannot be odd.

Q9. What is $[x] + [- x]$ ?

This equals $0$ if $x$ is an integer and $- 1$ otherwise. For example $[3] + [- 3] = 3 - 3 = 0$ ; but $[2.4] + [- 2.4] = 2 + (- 3) = - 1$ . It is a common JEE identity.

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1. Greatest Integer Function

Key Properties of [x]

2. Fractional Part Function

Key Properties of {x}

3. Even and Odd Functions

Key Properties

4. Periodic Functions

How to Test for Periodicity

Key Properties

Common Mistakes to Avoid

Frequently Asked Questions

Q1. What is the greatest integer function?

Q2. What is the fractional part of a number?

Q3. How do I check if a function is even or odd?

Q4. Can a function be both even and odd?

Q5. What is the period of sin(ax+b)?

Q6. How do I find the period of a sum of periodic functions?

Q7. Is every constant function periodic?

Q8. Why is f(0)=0 for an odd function?

Q9. What is [x]+[−x]?

Ready to master Functions?

Key Properties of $[x]$

Key Properties of ${x}$

Q5. What is the period of $sin (a x + b)$ ?

Q8. Why is $f (0) = 0$ for an odd function?

Q9. What is $[x] + [- x]$ ?