Function

Definition of a Function

A function is a specific type of relationship or rule that assigns each input value to exactly one output value.

Notation

Function is denoted as f(x) and read as f of x, where f is the name of the function, f(x) gives value on y axis or y in general. In the function f(x)= 2x, 2x is the rule to calculate f(x).

Domain and Range

Domain is set of possible values of input i.e. x and x axis. Range is set of possible values of output i.e. f(x),y and y axis.

Graph of a Function

For f(x)=2x

First we will take 3 values such that x and f(x) i.e. y are easy to plot on graph

For x=1

f(x)=y= 2×x

f(1) =2 here the variable x is changed with value of x this does not change the function. f(1) can be used as value too.

For x=2

f(x)=2×2

f(2)=4

For x=3

f(x)=2×3

f(3)=6

and then we will plot the points which we got as (1,2),(2,4),(3,6).

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Polynomial function

Any function which has no negative integer power of x,There is no non-real coefficient of x and the cofficient of highest degree of x is not zero is called Polynomial Function.

Definition-f(x)=a₀+a₁x+a₂x²...a_nxⁿ, here n can't be negative and only inteager is allowed.

Behaviour-

Domain:the domain is (-infinity,infinity).

Graph: smooth and the term with highest degree of x decides the graph.

example-f(x)=x²

Rational function

Any function which can be written as p(x)/q(x) and q(x)≠0 just like rational number which are written in form of p/q.

Definition:A rational function is a fraction of two polynomial function,where denominator is not zero.

Behaviour-

Domain:All real numbers except where q(x)=0.

Holes:occurs if there is a factor z is present in both p(x) and q(x).

Example-F(x)=1/x

Exponential function

Any function which has power as x and base is not less than 0,not equal to 1 and not multiplied by 0.

Definition-An exponential function is a mathematical formula of the formf(x)=a.b^x,where variable x is a exponent,b is base(b>0,≠1) and a is the initial value(a≠0).

Behaviour-

Domain and Range:Domain is (-infinity,infinity) for all real numbers.Range is (0,infinity) for all positive real numbers

Example-f(x)=2^x

Piecewise Function

Any function which has more than one rule.

Example-f(x)=x² if x>=0

f(x)=-x if x<0