Section 3.4 Library of Functions; Piecewise-defined Functions.
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Transcript of Section 3.4 Library of Functions; Piecewise-defined Functions.
Constant Function ( )f x bx y=b
-3
-2
-1
0
1
2
3
Dom ( , )f
Ran f b
-intercept
none if 0
x
b
-intercept
y
y b
( ) is an even function.y f x b
Identity Function ( )f x xx y=x
-3
-2
-1
0
1
2
3
Dom ( , )f
Ran ( , )f
-intercept
0
x
x
-intercept
0
y
y
( ) is an odd function.y f x x
Square Function 2( )f x xx y=x2
-3
-2
-1
0
1
2
3
Dom ( , )f
Ran [0, )f
-intercept
0
x
x
-intercept
0
y
y
2( ) is an even function.y f x x
Cube Function 3( )f x xx y=x3
-3
-2
-1
0
1
2
3
Dom ( , )f
Ran ( , )f
-intercept
0
x
x
-intercept
0
y
y
3( ) is an odd function.y f x x
Square Root Function ( )f x xx y=x1/2
0
1
2
3
4
7
9
Dom [0, )f
Ran [0, )f
-intercept
0
x
x
-intercept
0
y
y
( ) is neither odd nor even.y f x x
Cube Root Function 3( )f x xx y=x1/3
-27
-8
-1
0
1
8
27
Dom ( , )f
Ran ( , )f
-intercept
0
x
x
-intercept
0
y
y
3( ) is an odd function.y f x x
Reciprocal Function ( ) 1f x xx y=1/x
4
2
1
1/2
1/3
1/4
0
Dom
( ,0) (0, )
f
Ran
( ,0) (0, )
f
-intercept
none
x
-intercept
none
y
1( ) is an odd function.y f x
x
( )f x xx y=|x|
-3
-2
-1
0
1
2
3
Absolute Value Function
Ran [0, )f
-intercept
0
x
x
-intercept
0
y
y
( ) is an even function.y f x x
Dom ( , )f
Greatest Integer Functionx y= x
-1
-0.5
0
0.5
1
1.5
2
Dom ( , )f
Ran f
-intercept
0
x
x
-intercept
0
y
y
( ) is neither odd nor even.y f x x « ®
expression 1 if condition 1
expression 2 if condition 2( )
expression n if condition
f x
n
Use when x values satisfy condition n
Use when x values satisfy condition 1
Sometimes we need more than one formula to specify a function algebraically. In this case the formula used to evaluate the function depends on the value of x.
Piecewise Defined Functions
The following is a quick example of a piecewise defined function
2
32 5.5 if 2( )
13.8 2.5 if 2
x xf x
x x
(1) 32 5.5(1)f 2(4) 13.8 2.5(4)f
= 26.5
= 53.8
Use when x values are greater than 2
Use when x values are less than or equal to 2
Notice
Example 1
2
32 5.5 if 2( )
13.8 2.5 if 2
x xf x
x x
Notice that the domain of f , in this case, is the set all real numbers. That is, Dom f = (– , )
The following is a quick example of a piecewise defined function
Example 1
The percentage p (t) of buyers of new cars who used the Internet for research or purchase since 1997 is given by the following function.† (t = 0 represents 1997).
10 15 if 0 1( )
15 10 if 1 4
t tp t
t t
Notice that the domain of p is the interval [0 , 4]. That is, Dom p = [0 , 4].
†The model is based on data through 2000. Source: J.D. Power Associates/The New York Times, January 25, 2000, p. C1
Example 2
10 15 if 0 1( )
15 10 if 1 4
t tp t
t t
This notation tells us that we use the first formula, 10t + 15, if 0 t < 1, or, t is in [0, 1)
the second formula, 15t + 10, if 1 t 4, or, t is in [1,4]
Example 2
10 15 if 0 1( )
15 10 if 1 4
t tp t
t t
Thus, for instance, p(0.5) = 10(0.5) + 15 = 20 Here we used the first formula since
0 0.5 < 1, or, equivalently, 0.5 is in [0, 1).
p(2) = 15(2) + 10 = 40 We used the second formula since 1 2 4, or equivalently, 2 is in [1,
4]. p(4.1) is undefined p (t ) is only defined if 0 t 4.
Example 2