(MTH 250)

Lecture 2

Calculus

Previous Lecture’s Summary

• Introduction.

• Purpose of Calculus.

• Axioms of Order.

• Absolute value.

• Archimedean Property.

• Axioms of Archimedean Field.

• Axiom of Completeness.

• Completeness of R.

Today's Lecture

• Functions

• Graph of functions

• Domain & range of functions

• Piecewise defined functions

• Increasing & decreasing functions

• Maxima & Minima

• Even & odd functions

• Translation, reflection and dilatation

FunctionsFunctions

• A function f from set A to set B is a rule of correspondence that assigns to each element x in set A exactly one element y in set B.

A B

• A is called the domain (or set of allowable inputs) of f• B is called the co-domain of f and contains • Range (or set of outputs) of f.

f

x y =f(x)

• Each element in A (domain) must be matched with an element of B (range).

• Each element in A is matched to not more than one element in B.

• Some elements in B may not be matched with any element in A.

• Two or more elements of A may be matched with the same element of B.

Cont..Cont..

ContCont.

• Functions

x

Domain

y1

y2

Range

f

y

Range

fx1

Domain

x2

• Not a function

Cont..Cont..

x

Domain

y1

y2

Range

f

y

Range

fx1

Domain

x2

Cont..Cont..

• For a given input x, the output of a function f is called the value of f at x or the image of x under f..

• Sometimes we write y= f(x).

• The input variable x is called the independent variable and the output y is called the dependent variable.

• For the function f(x) = 3x - 4, f(5) = 3(5) - 4 = 15 - 4 = 11, f(-2) = 3(-2) - 4 = - 6 - 4 = -10.

• We say, f of 5 is 11, and, f of -2 is -10.

Graph of Functions

• We call a function f as real-valued function of a real variable, if both independent and dependent variables are real numbers.

• The graph of a real-valued function f in the xy-plane is defined by the graph of equation y=f(x).

• The points on the graph of the function f(x) are of the form (x , f(x)) or (x , y) where y=f(x).

Cont.. Cont..

Graph of y = |x|

22

11

00

11

22

|x|x

Cont.. Cont..

Graph of y=x²

4

93

11

00

11

42

y=x²x

2

Cont.. Cont..

Graph of

4

83

21

10

0.51

0.252

x

-4 -3 -2 -1 1 2 3 4 5

-30

-20

-10

10

20

30

x

yxy 2

2

xy 2

Cont.. Cont..

• Vertical line test: A curve in xy-plane is the graph of

some function f if and only if no vertical line intersects the curve more than once. For example following curve is a function

Domain & range of functionsDomain & range of functions

• Generally, the domain is implied to be the set of all real numbers that yield a real number functional value (in the range).

• Some restrictions to domain:

1. Denominator cannot equal zero.

2. Radicand must be greater than or equal to zero.

3. Practical problems may limit domain.

Cont..Cont..Finding the Range of a function:Draw a graph of the function for its given domain. The range is the set of values on the y-axis for which a horizontal line drawn through that point would cut the graph.

Cont..Cont..

f(x) = (x-2)2 +3, xR

The Range is f(x) ≥ 3 Range

Domain

Cont..Cont..

f(x) = 3 – 2x , xR

The Range is f(x) < 3.

Cont..Cont..

Piecewise defined functionsPiecewise defined functions

• A piecewise defined function is a function which assumes different values in different pieces (or parts).

• For example

01

012)( 2 xifx

xifxxf

Cont..Cont..

Cont..Cont..

f(x) = 3, for x 0

f(x)= 3+ x2, for 0< x 2

( ) 1 for 22

xf x x

Increasing & Decreasing FunctionsIncreasing & Decreasing Functions

• Increasing: Graph goes “up” as you move from left to right.

• Decreasing: Graph goes “down” as you move from left to right.

• Constant: Graph remains horizontal as you move from left to right.

)()(, 2121 xfxfxx

)()(, 2121 xfxfxx

)()(, 2121 xfxfxx

Cont..Cont..

f(x) = 3, for x 0

f(x)= 3+ x2, for 0< x 2

( ) 1 for 22

xf x x

Cont..Cont..

Cont..Cont..

Exercise: Find the interval where y is increasing, decreasing or constant.

Maxima & minimaMaxima & minima

• Relative maximum is the point x where the function y= f(x) assume the maximum value over a given interval (has a peak)

• Relative minimum is the point x where the function y= f(x) assume its minimum value over a given interval (has a Valley)

• Global maximum is the point x where the function y= f(x) assume its overall maximum value through the domain.

• Global minimum is the point x where the function y= f(x) assume its overall minimum value through the domain.

Maxima & minimaMaxima & minima

Even & Odd functionsEven & Odd functions

• Even function: If you plug in –x instead of x in a function f(x) and you get the original function, then it’s even, i.e. f(-x) = f(x)

• Odd function: If you plug in –x instead of x in a function f(x) and you get negative of the original function, then it’s odd, i.e. f(-x) = -f(x).

Translation, Reflection & DilatationTranslation, Reflection & Dilatation

• Vertical Shifts: If k is a real number and f(x) is a function, the graph of y = f(x) + k is that of f(x) shifted vertically by k units. If k > 0 then the shift is upward and if k < 0, the shift is downward.

• Horizontal Shifts: If h is a real number and f(x) is a function, we say that the graph of y = f(x - h) is the graph of f(x) shifted horizontally by h units. If h<0, then the shift is right and if h>0, then the shift is left.

Cont.. (vertical translations)Cont.. (vertical translations)

Graph y = |x|

22

11

00

11

22

|x|x

Cont.. (vertical translations)Cont.. (vertical translations)

Graph y = |x| - 1

122

011

-100

011122

|x| -1|x|x

Cont.. (vertical translations)Cont.. (vertical translations)

Graph y = |x| +2

422

311

200

311422

|x| +2|x|x

Cont.. (vertical translations)Cont.. (vertical translations)

y = 3x2

y = 3x2 – 3

y = 3x2 + 2

Cont.. (horizontal translations)Cont.. (horizontal translations)

y = 3x2

y = 3(x+2)2

y = 3(x-2)2

Cont.. (summary of translations)Cont.. (summary of translations)

for c>0

To Graph: Shift the Graph ofy = f(x) by c units

y = f(x) + c UP

y = f(x) - c DOWN

y = f(x + c) LEFT

y = f(x - c) RIGHT

Cont.. (reflections)Cont.. (reflections)

• Reflection across the x-axis: The graph of the function, y = - f(x) is the reflection of the graph of the function y = f(x).

• Reflection across the y-axis: The graph of the function, y = f(-x) is the reflection of the graph of the function y = f(x).

Cont.. (reflections across x-axis)Cont.. (reflections across x-axis)

f(x) = x3

f(x) = -x3

Cont.. (reflections across y-axis)Cont.. (reflections across y-axis)

f(x) = x3

f(-x) = (-x)3 = -x3

Cont.. (stretching and shrinking)Cont.. (stretching and shrinking)

• Vertical Stretching and Shrinking: The graph of

y = af(x)

is obtained from the graph of y = f(x) by

1. stretching the graph of y = f ( x) by a when a > 1, or

2. shrinking the graph of y = f ( x) by a when 0 < a < 1.

Cont.. (vertical stretching & shrinking)Cont.. (vertical stretching & shrinking)

y = |x|

y = 0.5|x|

y = 3|x|

Lecture Summary Lecture Summary

• Functions

• Graph of functions

• Domain & range of functions

• Piecewise defined functions

• Increasing & decreasing functions

• Maxima & Minima

• Even & odd functions

• Translation, reflection and dilatation