Functions-Types of Function

What is Function in Math?

A function is a rule that relates each element of a set X with exactly one element of set Y. It also is defined as the relationship between one variable (the independent variable) and another variable (the dependent variable).

• “…each element…” means that every element in set X is associated with or connected to at least one element in set Y.

(Note that there may be some elements in Y that remain unrelated to any elements in X, and that’s acceptable.)

• “…exactly one…” means that a function is single-valued means that there is only one output of X.

The formula for the area of a circle, A = πr², gives the dependent variable A (the area) as a function of the independent variable r (the radius).

 Functions are fundamental building blocks in mathematics, and they have various applications in science, engineering, and computer science.

Representation of Functions

Let f be a function from X to Y is represented as                       

f :X Y                    

read as “y is a function of x”.

Functions are generally represented as y = f(x).

Let, f(x) = x³.

It is said as f of x is equal to x cube.

Functions can also be represented by g(), t(),… etc.

The domain of a Function

The domain of a function is the set of all possible inputs for the function.

The codomain of a Function

The codomain or set of destinations of a function is the set into which all of the output of the function is constrained to fall.

Range of a Function

The range of a function refers to all the possible outputs of a function.

Example:

Let f(x) = x². Find the domain and range of f.

 Solution:

                  f(x) is defined for every real number x. Further for every real number x, f(x) = x² is a non-negative real number.

 So, Domain f = Set of all real numbers.

 Range f = Set of all non-negative real numbers.

Vertical line test

A vertical line test is used to determine whether a curve is a function or not. If any curve cuts a vertical line at more than one point, then the curve is not a function.

Types of Function

There are various types of functions given as:

• One-to-one Function (Injective Function)
• Onto Function (Surjective Function)
• One to One and Onto function (bijective function)
• Into Function
• Polynomial Function
• Linear Function
• Constant Function
• Identity Function
• Quadratic Function
• Rational Function
• Algebraic Functions
• Cubic Function
• Even and Odd Function
• Periodic Function
• Composite Function

Into Function

If a function f: A → B  is such that Ran f ⊂ B i.e., Ran f ≠ B, then f is said to be a function from A into B.

Fig f is a function. But Ran f ≠ B.

Therefore, f is a function from A to B.

 Onto (Surjective) function

If a function f: A→B is such that

Ran f = B i.e., every element of B is the image of some elements of A, then f is called an onto function or a surjective function.

One to One and Into (Injective) function

 If a function f from A into B is such that second elements of no two of its ordered pairs are equal, then it is called an injective (1 – 1, and into) function. The function shown in Fig is such a function.

One to One and Onto function (bijective function)

 If f is a function from A onto B such that the second element of no two of its ordered pairs are the same, then ƒ is said to be (1-1) function from A onto B. Such a function is also called a (1-1) Correspondence between A and B. It is

also called a bijective function. Fig shows a (1-1) correspondence between the sets A and B.

(a, z), (b, x) and (c, y) are the pairs of corresponding elements i.e., in this case

f= {(a, z), (b, x). (c, y)} which is a bijective function or (1-1) correspondence

between the sets A and B.

Polynomial Function

A polynomial function is a function that is defined by a polynomial expression in one or more variables. It consists of non-negative integer powers of the variable(s) multiplied by constants.

Example: f(x) = 3x^2 + 2x – 1 is a polynomial function of degree 2.

Linear Function

A linear function is a polynomial function of degree 1, and its graph is a straight line. It has the form f(x) = mx + b, where m is the slope and b is the y-intercept.

Example: f(x) = 2x + 3 is a linear function.

Constant Function

A constant function is a function that always returns the same constant value, regardless of the input.

Example: f(x) = 5 is a constant function as it always gives the output 5.

Identity Function

The identity function is a function that returns the same value as its input. The domain and codomain are the same for the identity function.

Example: f(x) = x is the identity function.

Quadratic Function

A quadratic function is a polynomial function of degree 2. Its graph is a parabola.

Example: f(x) = x^2 + 2x – 3 is a quadratic function.

Example

  A rough sketch of the functions

i) ((x, y) | 3x+y = 2)

ii) ((x, y) l y= ½ x²)

Solution

i) The equation defining the function is 3x + y = 2

⇒ y = -3x + 2

We know that this equation, being linear, represents a straight line. Therefore, for drawing its sketch or graph only two of its points are sufficient.

When

x=0, y = 2.

y = 0, x = 2/3 = 0.6 nearly.

 So two points on the line are A (0, 2) and B = (0.6, 0).

Joining A and B and producing the AB bar in both

directions, we obtain the line AB i.e., the graph of the given function.

ii) The equation defining the function is

y =1/2 x².

Corresponding to the values {…,-3, -2, -1, 0, 1, +2, +3} of x, values of y are 0, .5, 2, 4.5, …

We plot the points

(0, 0), (± 1, .5), (± 2, 2),(± 3, 4.5),… Joining them by means of a smooth

curve and extending it upwards we get the required graph. We notice that:

i) The entire graph lies above the x-axis.

ii) Two equal and opposite values of x corresponding to every value of y (but not vice versa).

iii) 0 As x increases (numerically) y increases and there is no end to their increase. Thus the graph goes infinitely upwards.

Rational Function

A rational function is a function that can be represented as the ratio of two polynomial functions.

Example: f(x) = (3x^2 + 2x – 1) / (x – 2) is a rational function.

Algebraic Functions

An algebraic function is a function that can be constructed using algebraic operations (addition, subtraction, multiplication, division) and applying roots and powers to the variable(s).

Example: f(x) = √(x^3) + 2x is an algebraic function.

Cubic Function

A cubic function is a polynomial function of degree 3. Its graph often has an S-shape.

Example: f(x) = x^3 – 4x^2 + 2x + 1 is a cubic function.

Even and Odd Function

An even function is a function that satisfies f(x) = f(-x) for all x in its domain.

 An odd function satisfies f(x) = -f(-x) for all x in its domain.

Example of an even function: f(x) = x^2

Example of an odd function: f(x) = x^3

Periodic Function

A periodic function is a function that repeats its values at regular intervals. There exists a positive constant ‘P’ such that f(x + P) = f(x) for all x in the domain.

Example: f(x) = sin(x) is a periodic function with a period of 2π.

Composite Function

A composite function is formed by applying one function to the output of another function.

Example: If f(x) = 2x and g(x) = x + 3, then the composite function (g ∘ f)(x) = g(f(x)) = g(2x) = 2x + 3.

Set-Builder Notation for a function

 We know that sub-builder notation is more

suitable for infinite sets. So is the case in respect of a function comprising an infinite

number of ordered pairs. Consider, for instance, the function

f = {(1,1), (2, 4), (3, 9), (4, 16)….}

Dom f= {1, 2, 3, 4,…}.

Ran f = {1,4,9, 16, …}

This function may be written as:

f = {(x, y) l y =r², x ∈ N}

For the sake of brevity this function may be written as:

f = function defined by the equation y=x², x ∈ N.

Or, to be still more brief: The function, x², x ∈ N.

In algebra and Calculus the domain of most functions is R and if evident from the context it is, generally, omitted.

 The inverse of a function

If a relation or a function is given in the tabular form i.e., as a set of ordered pairs, its inverse is obtained by interchanging the components of each ordered pair.

The inverse of r and f is denoted r^-1 and f^-1 respectively.

If r or f are given in set-builder notation the inverse of each is obtained by interchanging x and y in the defining equation. The inverse of a function may or may not be a function.

The inverse of the linear function

{(x, y) y= mx+ c) is ((x, y) l x= my+ c) which is also a linear function.

Briefly, we may say that the inverse of a line is a line.

The line y= x is clearly self-inverse. The function defined by this equation i.e., the function (x, y) | y= x) is called the identity function.

Example

 Find the inverse of

i) {(1, 1), (2, 4). (3, 9), (4, 16), x ∈ Z⁺}

ii) {(x, y) l y= 2x+ 3, x ∈ R)

Which of these are functions.

Solution

i) The inverse is:

{(2, 1), (4, 2), (9,3), (16, 4)…}

This is also a function.

Note: Remember that the equation

y= √x, x ≥ 0

defines a function but the equation y² = x, x ≥ 0 does not define a function.

The function defined by the equation.

y= √x, x ≥ 0

 is called the square root function.

The equation y² = x ⇒ y = ±√x

Therefore, the equation y= x (x ≥ 0) may be regarded as defining the union of the functions defined by

y = √x, x ≥ 0 and y = -√x, x ≥ 0.

ii) The given function is a linear function.

Its inverse is: {(x, y) | x= 2y + 3}

Which is also a linear function.

Points (0, 3), (-1.5, 0) lie on the given line and points (3, 0), (0, -1.5) lie on its inverse.