Study Notes for CAT 2021: Functions

By Gaurav Gupta|Updated : May 22nd, 2021

Functions

Introduction:

Functions provide us with a convenient way to handle the relationship between the values of one variable quantity that depends on the values of another variable quantity. For example, let us image a spherical rubber balloon into which air is being pumped. The radius o the balloon (r) is changing with time t. in mathematics, we say that r is a function of time t and symbolically, it may be written,

Functions

Introduction:

Functions provide us with a convenient way to handle the relationship between the values of one variable quantity that depends on the values of another variable quantity. For example, let us image a spherical rubber balloon into which air is being pumped. The radius o the balloon (r) is changing with time t. in mathematics, we say that r is a function of time t and symbolically, it may be written,

r=f(t): r is function of time t.

Similarly the volume of the balloon also depends on time t. Hence we can write

r=g(t): r is function of time t.

Different letters f and g are used because they represent different mathematical functions.

In general, if the values of a variable y depend on the values of another variable x, we write

y=f(x) i.e., y is a function of x.

Note: f(x) will obviously represent some expression of x.

Independent and Dependent Variables:

In the function y=f(x) is an expression in terms of x. Here x is known as the independent variable and y (whose values depend on x) is known as dependent variable.

Domain and Range:

While defining real-valued functions, we have to observe some restriction. One such restriction is that we can never divide by zero (0). Hence in the function

image001, x cannot be equal to 1.

The set of values which x can take (so that the function is well defined) is known as the set of domain for that function.

Similarly the dependent variable y in the function y=f(x), cannot take real values always. For example, in the function y=sinx, the variable y cannot take values which are greater than 1 or less than -1.

The set of values which y can take is known as the Range of the function y=f(x)

Intervals and Notations:

The domain and the range for different functions are the subsets of real numbers which are called intervals.

Open Interval:

If x can take values which lie strictly between a and b, then we can write

image003

Closed Interval:

If x can take values which lie strictly between a and b or equal to a or equal to b, then we can write

image004

Half-open Interval:

If only one end point is included for values of x, then the interval is called as half-open interval

image005

Infinite Interval:

image006

Important Functions:

  • Absolute value function (modulus function)

f(x)=|x| = magnitude of x or the positive value of x.

The expression |x| can be further split as follows:

image007

image008

Continuity

The graph of y=|x| is continuous (i.e. no break in the curve) but has a corner at origin as shown.

Domain & Range

The Domain of the function f(x) is x∈R and Range is image009

  • Greatest integer function (unit step function):

y=[x]= the greatest integer less than or equal to x.

It can also be simplified as:

y=[x]=n if n≤x < n+1 where n is an integer.

Continuity

The graph of f(x) is discontinuous (i.e. break in the curve) at integral value of x.

image010

Domain & Range

The Domain of the function f(x) is x∈R and Range is y∈R.

  • Signum Function:

image011

 

This can also be written as

image012

image013

Continuity

The graph of f(x) is continuous for all values of x except at x=0 where there is a break in the curve.

Domain & Range

The Domain of the function f(x) is x∈R and Range is set {-1, 0, 1}

  • Logarithmic Function:

image014

If a>1, y increases as x increases (as seen from graph).

image015

If 0<a<1, decreases as x increases.

Continuity

The graph of f(x) is continuous (i.e. break in the curve) in the respective domain.

Domain & Range

The Domain of the function f(x) is x>R and Range is y∈R.

Properties of logarithmic function:

image017

image018

 

  • Exponential Function:

image019

This function is the inverse of logarithmic function i.e. it can be obtained by interchanging x by y in image020

As observed from the graph, if a>1 then y increases as x increases.

If 0<a<1, then y decreases as x increases.

image021

Continuity

The graph of f(x) is continuous (i.e. no break in the curve) everywhere.

image022

Domain & Range

The Domain of the function f(x) is x∈R and Range is y>0.

  • Trigonometric Functions:

image023

Period

Period of image025

Period of image026

Continuity

The graph of image028 is continuous (i.e. no break in the curve) everywhere.

image024

image027

The graph of image029 is discontinuous (i.e. break in the curve) at image030

Domain & Range

The Domain of image031 is x∈R and Range is set [-A, A].

The Domain of image033 and Range is y∈R.

image032

  • Inverse Trigonometric Functions:

image034

image035

As observed from graph,

image036 increases as x increases,

and image037 decreases as x increases

and image038 increases as x increases.

image040

Continuity

The graphs image039 are continuous (i.e. no break in the curve) everywhere in their respective domains.

image001

Domain & Range

f(x)=sin-1x: The Domain of f(x) is set [-1, 1] and the Range is set [-π/2, π/2].

f(x)=cos-1x: The Domain of f(x) is set [-1, 1] and the Range is set [0, π].

f(x)=tan-1x: The Domain of f(x) is set (-∞, ∞) and the Range is set (-π/2, π/2).

 

Odd, Even and Periodic Functions:

  1. Even Function

If a function f(x) is symmetrical about Y-axis i.e. f(-x)=f(x) for all values of x, then the function y=f(x) is called an even function.

  1. Odd Function

If a function f(x) is symmetrical about origin i.e. f(x)=-f(-x) for all values of x, then the function y=f(x) is called an odd function.

  1. Periodic Function

If there exists a constant T such that f(x+T)=f(x) is satisfied by all values of x, then the function y=f(x) is called a periodic function.

The smallest positive value of T satisfying (f(x+T)=f(x) is known as the period of the function f(x).

Comments

write a comment

Follow us for latest updates