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8/12/2019 Lesson 2_Graphs, Piecewise,Absolute,And Greatest Integer
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FUNCTIONS
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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).
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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
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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)
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DEFINITION: PIECEWISE DEFINED FUNCTION
if x
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if x
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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
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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
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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
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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
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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|.