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

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

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) = x^{2
}, then f is one-to-one function, and g is not.
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) = n
^{2
} - 2

- g: R → R by f(n) = 3 n - 4

- Then, f is onto, and g is not.

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.

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

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) = 4x^{3
} – 3x^{2
}+ 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 is denoted by ,
thus, f (x) = sgn (x)
Where,

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

*
*

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

Properties

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*

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*

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*

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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 = log
_{b
}x is the inverse function of exponential function that is b^{y
} = x.