Chapter 1 -Part 1 Real Functions

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Chapter 1Calculus of One VariableEEM1016 Engineering Mathematics I

Trimester 2 Session 2013/14

Prepared by:Nasrin Sadeghianpour

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1. Real functions of a real variable2. Limits and continuity3. Differentiation & its applications4. Integration & its applications

Outline of Chapter 1

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Part 1 Real functions of a real variable

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If a variable y depends on variable x in such a way that each value of x determines exactly one value of y, then we say that y is a function of x.

Two commonly used methods of representing functions are:

1. By formulas 2. By graph

Definition of a function

)(xfy

Dependant variable Independent

variable (argument)

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Let X and Y be any nonempty sets. A function or mapping f from X to Y denoted by is a rule that assigns to each element of Y exactly one element of X. We say that X is the domain of f.

We write or to indicate that the element is the value assigned by the function to element In this case, we say that y is the image of x. The set of all images is called the range or image set of f, denoted by

Another definition of a function

YXf :

y f x ( )

( ) f x x X

y Y

R f .

YXf :

x X .f

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Example 1X:0123

Y:0149

)(xfy YXf :

Domain

Range2xy

Note that if the domain and the range of a function are both real, then is called a real-valued function of a real variable, or simply a real function.

fYXf :

In order to specify a function completely, the domain must be stated explicitly. Otherwise the domain is taken as the largest possible subset of R for which the real-valued function can be defined.

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Example 2

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Example 2 cont….

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Let be a function. The graph of is the set consisting of all points in the Cartesian coordinate plane, for all , i.e., the graph is the set

Graphs of Functionsf f

( , ( ))x f x,fx D

. )( and ),( xfyDxyx f

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Not every curve is the graph of a function! A curve in the xy-plane is the graph of a function

when it satisfies the vertical line property: any vertical line (a line parallel to the y-axis) intersects the curve at most once.

Graph of functions- Vertical line test

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Example 3

Solution (a) y is a function of x

Solution (b) y is a function of x

Solution (c) y is not a function of x

Solution (d) y is not a function of x

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Odd & Even functions

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Exercise

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Exercise

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Arithmetic Operations on functions

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Example 4

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Composition of functions

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Example 5

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Monotone functions Let A be a subset of R and let f be a

function. We say that is increasing on A if for all

such that decreasing on A if for all

such that

Some properties of functions

( ) ( )f x f y ,x y A;x y

( ) ( )f x f y ,x y A;x y

)(xf

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One-to-One or Injective Functions

Some properties of functions

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Some properties of functions

Inverse function

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Example 6

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Condition for existence of inverse

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How to find an inverse of a function

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Example 7

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Elementary functions

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Elementary functions

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Elementary functions

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Elementary functions

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Elementary functions

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Elementary functions

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Elementary functions

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Elementary functions

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Elementary functions

The tangent line to this

graph at (0.1) has slope 1.

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Elementary functions

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Elementary functions

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Elementary functions

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Elementary functions

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Elementary functions

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Elementary functions