MathematicsClass 11CBSE

Why a Vending Machine Behaves Like a Function but a Lucky Dip Does Not

Decide whether a relation is a function, learn the domain, range and graph of the identity, constant, polynomial, rational, modulus, signum and greatest-integer functions, find domains and ranges from formulas, and combine real functions.

What makes a relation a function?

A function from

A

B

is a relation in which **every element of

A

has exactly one image in

B

.** It is written

f : A \to B

, and the image of

x

f (x)

.

Press a button on a vending machine and you always get the same item — that is function-like behaviour. A lucky dip, where the same action can give different prizes, is not.

This part covers the function test, standard functions, domains and ranges from formulas, and combining functions.

How do you decide whether a relation is a function, including from its graph?

A relation is a function when no element of the domain has more than one image; on a graph, no vertical line may cut the curve more than once.

Worked examples — sets of pairs.

-

{(1, 2), (2, 3), (3, 4)}

— a function
-

{(1, 2), (1, 3), (2, 4)}

— not a function, since

1

has two images
-

{(1, 5), (2, 5), (3, 5)}

— a function; different inputs may share an output

Worked examples — graphs.

- **

y = x^{2}

— every vertical line meets it once, so it is a function
-

x^{2} + y^{2} = 25

** — the line

x = 3

meets it at

(3, 4)

and

(3, - 4)

, so it is not a function of

x

- **

x = y^{2}

** —

x = 4

gives

y = 2

and

y = - 2

, so not a function of

x

An everyday example. On a vending machine, two buttons may give the same packet of biscuits, which is allowed, but one button must never give two different items.

The substance. Many-one is fine; one-many is not.

What are the domain, range and graph of the identity, constant, polynomial, rational, modulus, signum and greatest-integer functions?

Each standard real function has a characteristic domain, range and graph shape, which you should be able to sketch and state from memory.

- Identity

f (x) = x

: domain

R

, range

R

; the line

y = x

- Constant

f (x) = c

: domain

R

, range

{c}

; a horizontal line
- Polynomial, e.g.

f (x) = x^{2}

: domain

R

, range

[0, \infty)

; a parabola
- Rational, e.g.

f (x) = \frac{1}{x}

: domain and range

R - {0}

; a hyperbola
- Modulus

f (x) = ∣ x ∣

: domain

R

, range

[0, \infty)

; a V shape
- Signum:

sgn x = ⎩ ⎨ ⎧ 1, 0, - 1, x > 0 x = 0 x < 0 range {- 1, 0, 1}

- Greatest integer

f (x) = [x]

, the greatest integer not exceeding

x

: domain

R

, range

Z

; a staircase

Worked examples.

[2.7] = 2 [5] = 5 [- 2.7] = - 3 sgn (- 4) = - 1 ∣ - 4∣ = 4

An everyday example. Age written in completed years on a form is the greatest integer function: someone aged

15.8

years writes

15

.

The trap. **

[- 2.7] = - 3

, not

- 2

**, because

- 3

is the greatest integer that does not exceed

- 2.7

How do you find the domain and range of a real function given by a formula?

**For the domain, exclude values that make a denominator zero or make an expression under a square root negative; for the range, solve

y = f (x)

for

x

and see which values of

y

are allowed.

Domains:**

f (x) = \frac{x ^{2} + 1}{x - 3} : R - {3} f (x) = \frac{1}{x ^{2} - 4} : R - {- 2, 2}

f (x) = x - 2 : [2, \infty) f (x) = \frac{1}{x - 1} : (1, \infty)

Domain and range together. For

f (x) = 9 - x^{2}

, we need

9 - x^{2} \geq 0

, so the domain is

[- 3, 3]

and the range is

[0, 3]

.

**Range by solving for

x

.** For

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

y (x - 3) = 2 x + 1 x (y - 2) = 1 + 3 y x = \frac{1 + 3 y}{y - 2}

y \neq = 2

, and the range is

R - {2}

.

Worked example.

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

has range

[2, \infty)

, since

x^{2} \geq 0

.

An everyday example. A formula for the side of a square plot from its area,

s = A

, only makes sense for

A \geq 0

— its domain is set by reality.

The substance. A square root in a denominator needs a strictly positive argument, which is why

(1, \infty)

is open at

1

How do you add, subtract, multiply and divide real functions, and what is the domain of the result?

Combine the formulas point by point, and take the domain as the intersection of the two domains, also removing any values that make the divisor function zero.

Worked example 1.

f (x) = x + 2

and

g (x) = x^{2} - 4

, both with domain

R

(f + g) (x) = x^{2} + x - 2 (f - g) (x) = - x^{2} + x + 6

(f g) (x) = (x + 2) (x^{2} - 4) = x^{3} + 2 x^{2} - 4 x - 8

(\frac{f}{g}) (x) = \frac{x + 2}{x ^{2} - 4} = \frac{1}{x - 2}, domain R - {- 2, 2}

Worked example 2.

f (x) = x - 1

with domain

[1, \infty)

, and

g (x) = 5 - x

with domain

(- \infty, 5]

domain of f + g = [1, 5] domain of \frac{f}{g} = [1, 5)

(f + g) (3) = 2 + 2 = 22

An everyday example. A tailor's total monthly cost is a fixed-cost function plus a cost that depends on the number of garments — the sum of two functions.

The substance. Cancelling does not restore excluded values:

\frac{f}{g}

above is still undefined at

x = - 2

Exam tip

What earns full marks on functions?

State domain and range in correct interval or set notation, sketch standard graphs with key points marked, and show the conditions used to find a domain.

- Function test: every element has exactly one image
- Vertical line test for graphs
- Signum: values

1

0

- 1

; greatest integer: round down
- Domain: denominators

\neq = 0

, square-root arguments

\geq 0

- Range: solve

y = f (x)

for

x

- Combined functions: intersect domains; exclude zeros of the divisor

The trap. Writing

[- x] = - [x]

. **

[- 2.7] = - 3

while

- [2.7] = - 2

.**

Did you know

Why does multiplying the signum of x by the modulus of x give back x?

The modulus

∣ x ∣

keeps the size of a number and throws away its sign. The signum

sgn x

keeps only the sign.

Multiply them and you put the number back together:

sgn (- 4) \times ∣ - 4∣ = - 1 \times 4 = - 4 sgn (3) \times ∣3∣ = 1 \times 3 = 3

sgn x \cdot ∣ x ∣ = x

for every real

x

, including

0

. Every number is simply a sign times a size.

Exam relevance

How are domain, range and special functions tested in JEE Main?

Functions are part of the JEE Main unit Sets, Relations and Functions, continued in Class 12 Relations and Functions, and they underpin Limits, Continuity and Differentiability.

What gets asked. Domains and ranges of combined rational and square-root expressions, graphs of modulus and greatest-integer functions, and limits and continuity of functions involving

[x]

∣ x ∣

and

sgn x

, where the jumps and corners of their graphs matter.

Question types. Multiple-choice and numerical-value questions; JEE Advanced adds harder domain and range problems.

The trap that costs marks. **Treating

[- x]

- [x]

**, which fails for every non-integer.

Key takeaways

What must you be able to do from this part?

- Function: each input has exactly one output; vertical line test for graphs
- Standard functions: identity, constant, polynomial, rational, modulus, signum, greatest integer — with domains and ranges
- **

[2.7] = 2

[- 2.7] = - 3

**;

sgn x \in {- 1, 0, 1}

- Domains:

x - 2

gives

[2, \infty)

;

\frac{1}{x ^{2} - 4}

gives

R - {- 2, 2}

- Range of

\frac{2 x + 1}{x - 3}

R - {2}

- Combining functions: intersect domains and exclude zeros of the divisor

Sketch

y = [x]

from

x = - 2

x = 3

, marking which end of each step is a filled dot.

What makes a relation a function?

How do you decide whether a relation is a function, including from its graph?

What are the domain, range and graph of the identity, constant, polynomial, rational, modulus, signum and greatest-integer functions?

How do you find the domain and range of a real function given by a formula?

How do you add, subtract, multiply and divide real functions, and what is the domain of the result?

What earns full marks on functions?

Why does multiplying the signum of x by the modulus of x give back x?

How are domain, range and special functions tested in JEE Main?

What must you be able to do from this part?

Related study notes

Ready to put this into practice?