Transcript Slide 1
FUNCTIONS
GRAPHS OF FUNCTIONS; PIECEWISE
DEFINED FUNCTIONS; ABSOLUTE VALUE
FUNCTION; GREATEST INTEGER FUNCTION
OBJECTIVES:
• 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
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 graph of 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).
Knowledge of the standard forms of the special
curves discussed in Analytic Geometry such as lines
and conic sections is very helpful in sketching the
graph of a function. Functions other than these
curves can be graphed by point-plotting.
To facilitate the graphing of a function, the
following 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.
EXAMPLE:
A. Sketch the graph of the following functions and
determine the domain and range.
1. f ( x ) x
2
2.G ( x) 9 x
3.G ( x) x 4
2
x 3x 2
4.h( x)
x 1
2
5.h( x) 9 x
2
6.g ( x) 3 x 2
SOLUTIONS:
1. f ( x) x
2
2.F ( x) 9 x
3.G( x) x 2 4
(9, 0)
(0, 4)
D : ,
R : 0,
x 2 3x 2
4.h( x)
x 1
D : ,9
R : 0,
5.h( x) 9 x
(0, 3)
(-1, 1)
D : , except 1
R : , except 1
D : ,
(-3, 0)
R : 4,
2
6.g ( x) 3 x 2
(-2, 3)
(3, 0)
D : 3,3
R : 0 ,3
D : ,
R : 3,
When the graph of a function is given, one can
easily determine its domain and range.
Geometrically, the domain and range of a function
refer 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 the
independent variable in its domain. Geometrically
speaking, this means:
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.
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. Using
the vertical line test, a vertical line when drawn
within –3 x 3 intersects the curve at two points.
Refer to the figure below.
(0, 3)
(3, 0)
(-3, 0)
(0, -3)
DEFINITION: PIECEWISE DEFINED FUNCTION
Sometimes a function is defined by more than
one rule or by different formulas. This function is
called a piecewise define function.
A piecewise defined function is defined by different
formulas on different parts of its domain.
if x 0
x
2
x 2 if x<0
if 0 x 3
2
.
f
(
x
)
9
x
1.f ( x )
x 3 if x 1
x 1 if x 0
Example:
EXAMPLE:
A. Evaluate the piecewise function at the
indicated values.
x 2 if x<0
1.f ( x )
x 1 if x 0
f(-2), f(-1), f(0), f(1), f(2)
if x 0
3x
2.f ( x ) x 1 if 0 x 2
( x 2) 2 if x 2
f(-5), f(0), f(1), f(5)
EXAMPLE:
B. Define g(x) = |x| as a piecewise defined
function and evaluate g(-2), g(0) and g(2).
Solution:
From the definition of |x|,
x if x 0
g( x )
x if x 0
Therefore
g ( 2 ) ( 2 ) 2
g( 0 ) 0
g( 2 ) 2
EXAMPLE:
Sketch the graph of the following functions and
determine the domain and range.
x2
1 x 2
if x 1
3x 2 if x 1
2. f ( x) 2
if x 1
x
4
1.g ( x) 2
3
if
if
x 2
3. f ( x )
2 x 1
if x 1
if x 1
DEFINITION: ABSOLUTE VALUE FUNCTION
Recall that the absolute value or magnitude of
a real number is defined by
x,
x
x,
if
if
x0
x0
Properties of absolute value:
1. a a
A number and its negativehave thesame absolut e value
2. ab a b
T heabsolut e value of a product is theproduct of theabsolut e values
a
a
3.
, b 0 T heabsolut e value of a ratiois theratioof theabsolut e values
b b
4. a b a b T he triangleineaquality
The graph of the function f ( x) x can be obtained
by graphing the two parts of the equation
x,
y
x,
if x 0
if x 0
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.
Example:
Sketch the graph of the following functions and determine
the domain and range.
1.f ( x ) x 3x 1
2.f ( x ) 3 2 x 5
DEFINITION: GREATEST INTEGER FUNCTION
The greatest integer function is defined by
x greatest integer less than or equal to x
Example:
0 0
1.1 1
0.1 0
0.3 0
1.2 1
0.9
2
1
0
1
1.9 1
2
2.1 2
3.4 3
3.4 -4
0.9 -1
Graph of greatest integer function.
Sketch the graph of y x
x
2 x 1
1 x 0
0 x 1
1 x 2
2x3
y x
2
1
0
1
2
y
o
x
EXERCISES:
A. Given the following functions, determine the domain and
range, and sketch the graph:
1. H : y 4 x 3
2. F : y 1 x
x 1 if x 3
7. F : y
2 x 1 if x 3
x 2 2x 1
5. g : y
x 1
3 if x 1
8. f : y 1 if 1 x 2
3 if x 2
6. h : y 2 4 x 2
9. G : y 4 x
3. G : y 1 2 x
4. h : y x 2 3
x 3x 4x 9
y
x x 12x 3
2
10.
2
2
EXERCISES:
B.Compute the indicated values of the given functions.
a.
t 4
3 if
f ( x ) t 1 if 4 t 4
t if
t4
f(6), f(4),andf(16)
x 2 4 if x 3
c. F( x )
2 if x 3
2
F(4),F(0),F(3), and F
3
4 if x 2
h
(
x
)
b.
1 if 2 x 2
3 if
2x
h(3),h(2),h ,h(e2 ), andh(2)
2
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 - |x – 2|.