# Chapter 2 : Functions

### Topics covered in this snack-sized chapter:

#### Functions Definition arrow_upward

• A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
• The output of a function f corresponding to an input x is denoted by f(x) (read " f of x").
•

• Let X and Y be two nonempty sets of real numbers.
• A function from X into Y is a relation that associates with each element of X a unique element of Y.
• ##### Domain:
• The set of all possible input values. For  function f(x), x is called the Domain of the function.
• ##### Range:
• The set of all images of the elements of the domain is called the Range of the function.
• For each element x in X, the corresponding element y in Y is called the image of x.
• ##### Example of Function:
• Any sequence can be considered as a function defined on a set of integers.
• Example: Sequence 2, 5, 8, 11, 14,… can be defined as a function as follows:
•     f : n → 3n - 1

#### One-to-One Functions arrow_upward

• A one-to-one or injective function f from set X to set Y is a function such that each x in X is related to a different y in Y.
• F is one-to-one or injective if for all elements  . If  then . No  in the range is the image of more than one  in the domain.
• As shown in figure that, for every element of  there is exactly one value of.
• Example: f: R→R is defined by f(n) = 2n + 3, g: R → R is defined by f(x) = x2 , then f is one-to-one function, and g is not.

• #### Onto Functions arrow_upward

• A function f : XY is said to be onto or surjective if for every y in Y, there is an x in X, such that f(x) = y. All the elements in the Y are used.
• A function is onto if two or more different elements in the domain corresponds to the same element in the range.
• Examples:
• We define f: R → R by f(n) = n2 - 2
• g: R → R by f(n) = 3 n - 4
• Then, f is onto, and g is not.

#### One-to-One Correspondences arrow_upward

• A One-to-One correspondence or bijection from a set X to a set Y is a function  that is both one-to-one and onto.

• #### Independent Variable arrow_upward

• For a function y = f(x), the variable x is called the Independent Variable.
• Because it can be assigned any of the permissible numbers from the domain.
• The independent variable is also called the Argument of the function.

• #### Dependent Variable arrow_upward

• The variable y is called the Dependent Variable, because its value depends on x.

• #### Boolean Functions arrow_upward

• This is a function that returns a true (for 1) or false (for 0) value instead of a numeric value. It uses principal logical relations “and”, “or”, “not”, “if…then” etc.
• In Boolean function both range and domain consists of true and false.
• Example:

• #### Real Valued Function arrow_upward

• The function for which the domain and range are the subsets of the set of real number R is called a Real valued Function.
• Uniform definition
• If a function is defined as
• , we say that it is uniformly defined.
• Piecewise Definition
• If a function  assumes different forms in different subsets of, we say that it is piecewise defined.

#### Algebraic Functions arrow_upward

• Functions with values that are obtained by adding, subtracting, multiplying or dividing constants and independent variable or raising a variable to a power are known as Algebraic Functions.
• Algebraic functions are of three types.
• ##### Polynomial Function:
• If a function y = f(x) is given by:
• Where,  are real numbers (sometimes called the coefficients of the polynomial) and n is any non – negative integer, then f(x) is called a Polynomial Function in x.
• The degree of a polynomial is the highest power of x in its expression. The function f(x) = 0 is also a polynomial but we say that its degree is “undefined”.
• Example: f(x) = 4x3 – 3x2 + 2 is a polynomial function with degree 3.
• ##### Rational Functions:
• If a function  is given by
• Where  and  are polynomial functions, then f(x) is called a Rational Function in x.
• ##### Irrational Functions
• The algebraic functions containing one or more terms having non – integral rational powers of x are called Irrational Functions.

• #### Signum Function arrow_upward

• Signum function is denoted by ,
• thus, f (x) = sgn (x)
•      Where,

• Domain of sgn (x) = R
• Range of sgn (x) = {-1, 0, 1}

• #### Exponential Functions arrow_upward

• If a is any number such that a > 0 and a ≠ 1  then an exponential function is a function in the form, , where a is called the base and x can be any real number.
• The Exponential Function with base  is the following function from R to .
•

#### Logarithmic Functions arrow_upward

• If a is any number such that a > 0 and a ≠ 1 and x > 0 then, .
• Symbolically,
•

• Properties
•

#### Inverse Functions arrow_upward

• Suppose  is a one-to-one correspondence.
• Then, there is a function  defined as follows:
• Given any element in Y.   is the unique element x in X such that

F(x) = y.

• The function   is called the Inverse Function for F.
• Example:
• The logarithmic function with base  is the inverse of the exponential function with base b.That is logarithmic function y = logb x is the inverse function of exponential function that is by = x.

#### Thank You from Kimavi arrow_upward

• Please email us at Admin@Kimavi.com and help us improve this tutorial.

• Tutorials, Videos and Quizzes - Real Simple Education

humanSuccess = (education) => { `Life, Liberty, and the pursuit of Happiness` }