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Section 5.2B Power amd Piecewise Defined Functions
Definition: A function is a function of the form kxr, where k and r are
real numbers.
Some Definitions Involving Exponents:
1. If a 6= 0, a0 = 1, a�1 = 1a , and, in general, a�x = 1
ax .
2. a1/2 =pa, a1/3 = 3
pa, and, in general, a1/n = n
pa.
3. am/n = npam = ( n
pa)m
The domain of f(x) = npg(x) is all real numbers (�1,1) if n is odd. If n is even then to find
the domain of f(x) = np
g(x), you must solve g(x) � 0 for x.
Example 1: Find the domain of the following power functions:
a) f(x) = 3px2 � 1
a) f(x) =px� 1
b)4x� 24px� 1
power
3 is odd, so domain is (-o@
X - 120 Domain+ I + I
X Z / [@
Root function vs on the bottom
of the fraction , so solve × -1 > 0 for × ( Can't have * Obooytoathe)
¥1fiend
b)
px
x2 � 3x� 4
Example 2: Rationalize the numerator of the function
px+ 2� 5
x� 2
Example 3: Compute the following for the function defined below. Simplify your answers.
f(x) =px+ 5
a) f(x+ h)
b) f(x+ h)� f(x)
2 Spring 2018, Maya Johnson
Both : xzo &X¥4&x¥ - 1
Top first : xzo Ignore -1 since it 's < 0.
BottomlyIYEXYIEO 894k€( × - 4) =o or ( xtD=o
×=4 or X= - 1
Multiply by th Conjugate -7 LEE
~M
( €2 +5)
Middle terms =(F+z - 5)(F+z + 5) =×f+2XFt2) - (5×5) .
cannon the toptar Hats
- ¥5622575
=IIzk÷+⇒@← intent:; roots .
= pets =#@
. xthtt.it#
c)f(x+ h)� f(x)
h
Definition: Functions whose definition involve more than one rule are calledPiecewise Functions.
To graph, graph each rule over the appropriate portion of the domain.
Example 4, Absolute value function: The absolute value of a number x, denoted by |x|,is the distance from x to 0 on the real number line. The plot and definition of absolute value
function if given below.
x
y
|x| =(x x � 0
�x x < 0
Example 5: Write f(x) = |6x+ 7| as a piecewise defined function.
3 Spring 2018, Maya Johnson
=lFthF#)h
=
txnFsfxtsXxfhFtxF5LxFhsXFthtsHxFsXFFDhTxEsxiFs-E5xFsEIhEIFYIasjxEEtEsTiaatF@t6xt7l-6xt7when.t
720's
,16×+71 = - (6×+7) who ex +7<0
Solve for × in bolt.
ask '
6x±}>=q⇒ 6 ÷ .
.
-
' 46
Therefore 6×+7<0 ⇒ X <- 7/6
ftp.byfmitifxxIY#f
Example 6: Find the domain and sketch graph of the function
f(x) =
8<
:x+ 1 if x < 1
1� x if x > 1
Example 7: Evaluate f(�2), f(�1), and f(4) for the following piecewise defined function.
f(x) =
8<
:x+ 5 if x �1
x2 if x > �1
4 Spring 2018, Maya Johnson
× = I is not defined
So domain
2) +5 =Dftp..tl ) +5=40
fl 4) = (4) 2=160
Sketch a graph of the function.
Example 8: Find the domain of the following piecewise defined function.
f(x) =
8<
:
2
x2 � 9x+ 14if 0 x 4
4x+ 5 if x > 4
Example 9: An electricity company charges its customers a base rate of 10 a month, plus 6
cents per kilowatt-hour(kWh) for the first 1200 kWh and 7 cents per kWh for all usage over 1200
kWh. Express the monthly cost E as a function of the amount x of electricity used.
5 Spring 2018, Maya Johnson
TV
for 2-so Domain T
5×2-9×+14EEAYXIIEEO any× = 7 or X=Z × = 4 rs also not a problem
but x=7 is not in the range of OE x<- 4
Cost : 10 +.
obx if 0 a- ×± Koo
But ,of × 7 1200 , say 1201 ,
the first 1200 costs
10 +. 064204=82 ,
and the 1kWh over is.
07
t.FI?boxnxYi:oYiFF#