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Functions

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CAT 2025 Lesson : Functions - Domain, Codomain & Range

2.1 Domain, Co-domain and Range

Brief definitions of the three are as follows.

1

) Domain is the set of all possible inputs or

x

-values.

2

) Co-domain is the set of possible outputs or values of

f(x)

that is defined by us.

3

) Range is the set of outputs that are actually generated for the given inputs or domain. These are the actual values that

f(x)

can possible take.

The following diagram is a pictographic representation of

f(x) = |x|

, where

x

is an integer between

-2

and

2

(both inclusive). And, the co-domain has been defined as {

-2, -1, 0, 1, 2

}.

As

f(x)

is never negative, the range does not include

-2

and

-1

and contains only the non-negative integers.

Two other classifications of functions are as below.

1

Onto Functions: Every element in the co-domain is related to at least one value in the domain. In this case, the co-domain is the range.

2

Into Functions: There is at least one element in the co-domain that is not related to a value in the domain. In this case, while the range is a subset of the co-domain, it has fewer values than the co-domain.

Example 1

For

f(x) = x^2

,
(I) What is the domain?
(II) What is the range?
(III) What is the co- domain if this is an onto function?
(IV) What could be a co- domain if this is an into function?

Solution

Domain: As

x

can take any real number, the domain of

x

is the set of all real numbers.

Range: As

x^2

can never be negative, the range of

x

is the set of all non-negative real numbers.

Co-domain for an Onto Function: The co-domain here has to be the range, i.e. the set of all non-negative real numbers.

Co-domain for an Into Function: The co-domain should include at least one item other than the range. Therefore, this could be the set of all real numbers.

Reminder: As discussed earlier, one-to-many and many-to-many relationships are not functions. As shown earlier,

y^2 = x

y = \pm \sqrt{x}

, where

x \ge 0

is not a function as each

x

-value (other than

0

) results in

2 \space y

-values.

3. Common Types of Functions

Some common types of functions are given below, with examples.

Linear Functions are of the form $y = mx + c$ . This results in a straight line. Adjacent is the graphical representation of $f(x) = 0.5x - 2$ Type: One-to-One function Domain: Set of all real numbers Range: Set of all real numbers
Quadratic Functions have a U-shaped or an inverted-U shaped curve, as discussed in the Quadratic *Equations* lesson Adjacent is the graphical representation of $f(x) = x^2 - 4$ Type: Many-to-One function Domain: Set of all real numbers Range: Set of all real numbers greater than or equal to $-4$
In Cubic Functions, the highest power of the polynomial is $3$ . Adjacent is the graphical representation of $f(x) = x^3$ Type: One-to-One function Domain: Set of all real numbers Range: Set of all real number
In Bi-quadratic Functions, the highest power of the polynomial is $4$ . Adjacent is the graphical representation of $f(x) = x^4$ Type: Many-to-One function Domain: Set of Real numbers Range: Set of real numbers greater than or equal to $-1$ .
Rational Functions are those where the numerator and denominator contain a polynomial. The function is not defined when the denominator equals zero. Adjacent is the graphical representation of $f(x) = \dfrac{2}{x - 1}$ Type: One-to-One function Domain: Set of real numbers other than $+1$ . Range: Set of real numbers other than $0$ .
The Modulus Function of a number provides the non-negative value of the number. To define it as a function, $f(x) = \|x\| = x \space \text{if} \space x \space \ge 0$ $= -x \space \text{if} \space x < 0$ The above function is represented as $f(x) = \|x\|$ Type: Many-to-One function Domain: Set of all real numbers Range: Set of all non-negative real numbers

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