Lesson 2_Graphs, Piecewise,Absolute,And Greatest Integer

8/12/2019 Lesson 2_Graphs, Piecewise,Absolute,And Greatest Integer

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FUNCTIONS


2/21

GRAPHS OF FUNCTIONS; PIECEWISE

DEFINED FUNCTIONS; ABSOLUTE VALUE

FUNCTION; GREATEST INTEGER FUNCTIONOBJECTIVES: sketch the graph of a function;

determine the domain and range of a

function from its graph; and

identify whether a relation is a function or

not from its graph.

define piecewise defined functions;

evaluate piecewise defined functions;

define absolute value function; and

define greatest integer function


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As we mentioned in our previous lesson, a function

can be represented in different ways and one of which

is through a graph or its geometric representation.We also mentioned that a function may be

represented as the set of ordered pairs (x, y). That is

plotting the set of ordered pairs as points on the

rectangular coordinates system and joining them will

determine a curve called the graph of the function.

The graphof a function f consists of all points (x, y)

whose coordinates satisfy y = f(x), for all x in the

domain of f. The set of ordered pairs (x, y) may also be

represented by (x, f(x)) since y = f(x).


4/21

Knowledge of the standard forms of the special

curves discussed in Analytic Geometry such as lines

and conic sections is very helpful in sketching thegraph of a function. Functions other than these

curves can be graphed by point-plotting.

To facilitate the graphing of a function, thefollowing steps are suggested: Choose suitable values of x from the domain of a

function and

Construct a table of function values y = f(x) from the

given values of x.

Plot these points (x, y) from the table.

Connect the plotted points with a smooth curve.


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1

23

)(.4

4)(.3

9)(.2

)(.1

2

2

2

x

xx

xh

xxG

xxG

xxf

A. Sketch the graph of the following functions and

determine the domain and range.

EXAMPLE:

23)(.6

9)(.5 2

xxg

xxh


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SOLUTIONS:

(-3, 0) (3, 0)

(0, 3)

(-2, 3)

(9, 0)(0, 4)

(-1, 1)

2)(.1 xxf xxF 9)(.2 4)(.32 xxG

1

23)(.4

2

x

xxxh

29)(.5 xxh 23)(.6 xxg

,0:

,:

R

D

,0:

9,:

R

D

,4:

,:

R

D

1,:

1,:

exceptR

exceptD

3,0:R

3,3:D

,3:

,:

R

D


7/21

When the graph of a function is given, one can

easily determine its domain and range.

Geometrically, the domain and range of a functionrefer to all the x-coordinate and y-coordinate for

which the curve passes, respectively.

Recall that all relations are not functions. A

function is one that has a unique value of the

dependent variable for each value of theindependent variable in its domain. Geometrically

speaking, this means:


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Consider the relation defined as {(x, y)|x2+ y2= 9}.

When graphed, a circle is formed with center at

(0, 0) having a radius of 3 units. It is not a function

because for any x in the interval (-3, 3), two ordered

pairs have x as their first element. For example, both

(0, 3) and (0, -3) are elements of the relation. Usingthe vertical line test, a vertical line when drawn

within3 x 3 intersects the curve at two points.

Refer to the figure below.

A relation f is said to be a function if and only if, in its

graph, each vertical line cuts or touches the curve

at no more than one point.

This is called the vertical line test.


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(0, 3)

(3, 0)(-3, 0)

(0, -3)


10/21

DEFINITION: PIECEWISE DEFINED FUNCTION

if x


11/21

if x


12/21

B. Define g(x) = |x| as a piecewise defined

function and evaluate g(-2), g(0) and g(2).

EXAMPLE:

Solution:

From the definition of |x|,

0x

0x

if

if

x

x)x(g

2)2(g

0)0(g

2)2()2(g

Therefore


13/21

Sketch the graph of the following functions and

determine the domain and range.

EXAMPLE:

2

23)(.2

3

2

4

)(.1

x

xxf

xg

if

if

if

1

21

2

x

x

x

if

if

1

1

x

x

112

1)(.32

xifx

xifxxf


14/21

DEFINITION: ABSOLUTE VALUE FUNCTION

Recall that the absolute value or magnitudeof

a real number is defined by

Properties of absolute value:

0,

0,

xifx

xifxx

yineaqualittriangleThebaba.4

valuesabsolutetheofratiotheisratioaofvalueabsoluteThe0b,b

a

b

a.3

valuesabsolutetheofproducttheisproductaofvalueabsoluteThebaab.2

valueabsolutesamethehavenegativeitsandnumberAaa.1


15/21

The graph of the function can be obtained

by graphing the two parts of the equation

separately. Combining the two parts produces the V-shaped

graph. It may help to generate the graph of absolute value

function by expressing the function without using absolute

values.

xxf )(

0if,

0if,

xx

xx

y

Example:

Sketch the graph of the following functions and determine

the domain and range.

5x23)x(f.2

1x3x)x(f.1


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DEFINITION: GREATEST INTEGER FUNCTION

greatest integer less than or equal to x

The greatest integer function is defined by

x

Example: 0

1.0

3.0

9.0

1

1.1

2.1

9.1

2

1.2

4.3

4.3

9.0

0

0

0

0

1

1

1

1

2

2

3

-4

-1


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Graph of greatest integer function.

xySketch the graph of

x xy

1x2

0x1 1x0

2x1 3x2

210

12


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x

y

o

EXERCISES


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EXERCISES:

22

2

2

4:.6

1

12:.5

3:.4

21:.3

1:.2

34:.1

xyh

x

xxyg

xyh

xyG

xyF

xyH

312

943.10

4:.9

23

211

13

:.8

312

31:.7

2

22

xxx

xxxy

xyG

xif

xif

xif

yf

xifx

xifxyF

A. Given the following functions, determine the domain and

range, and sketch the graph:


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EXERCISES:

B.Compute the indicated values of the given functions.

4t

4t4

4t

if

if

if

t

1t

3

)x(f

)16(andf),4(f),6(f

a.

x2

2x2

2x

if

if

if

3

1

4

)x(hb.

c.

)2(hand),e(h,2

h),2(h),3(h 2

3x

3x

if

if

2

4x)x(F

2

3

2Fand),3(F),0(F),4(F


21/21

C. Define H(x) as a piecewise defined function and

evaluate H(1), H(2), H(3), H(0) and H(-2) given by,

H(x) = x - |x2|.

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Lesson 2_Graphs, Piecewise,Absolute,And Greatest Integer