Why is 89 and 93 (-infinity, infinity) and not [7/4,infinity) and [-7,infinity)? Solution 89. k(x) = (4x -7)1/3 The function k(x) takes all the values on the real number line as x takes various values on the real number line. Both the domain and the range of the function k(x) are R, the set of all real numbers, or in interval notation, (- , ). When x = 7/4, k(x) = 0 and when x < 7/4, k(x) is negative. A negative real number can have a third root. For example, the 3rd root of - 1 is -1 as (-1)3 = -1. Also, if n is a positive number, then (-n)1/3 = (- 1*n)1/3 = -1(n)1/3 which is real number. Hence the domain of k(x) is (- , ) and not [7/4, ). 93. F(x) = (x+7)1/3. The function F(x) takes all the values on the real number line as x takes various values on the real number line. Both the domain and the range of the function k(x) are R, the set of all real numbers, or in interval notation, (- , ). When x = -7, F(x) = 0 and when x < -7, F(x) is negative. A negative real number can have a third root. For example, the 3rd root of -1 is -1 as (-1)3 = -1. Also, if n is a positive number, then (-n)1/3 = (- 1*n)1/3 = -1(n)1/3 which is real number. Hence the domain of F(x) is(- , ) and not [-7, )..

1. Why must each network router interface belong to a different network?
Solution
Routers have multiple interfaces that are used to connect to multiple networks. Typically, the
interfaces connect to various types of networks, which means that different types of media and
connectors are required. Often a router will need to have different types of interfaces. For
example, a router usually has FastEthernet interfaces for connections to different LANs and
various types of WAN interfaces to connect a variety of serial links including T1, DSL and
ISDN. .
The outside interfaces on both routers must be on the same subnet for traffic to pass from one
router to the other router. Just like your network switch has to be on the same network as your
internal interface on the other side of your router in order for traffic to pass from the switch to
the router and then in turn, your workstations have to be on the same network as your switch in
order for traffic to pass from the workstation to the switch.
Generally each interface will have its own IP address, distinct and separate from that of any other
interface.So that router knows to use that interface for destinations with a next hop of the remote
router when using IP without any additonal configuraiton. This includes subinterfaces, which
arise when a physical interface has trunking capabilities, such as with 802.1q VLANs or frame
relay circuits.
In unusual configurations, the same IP address can be on more than one interface (bridged
interfaces), but the situations where this is desirable are fairly rare.
It is possible to put multiple IP addresses on the same interface (so-called "secondary
addressing"), but this practice is disrecommended for many reasons and should generally only
be used in the short term, usually as a transitional step to minimize disruption during a "live"
network restructuring.
An IP router has to have a minimum of 2 IP addresses, because its function is deciding on where
to send the packet next. A minimal configuration of TCP/IP network where you need a router is a
connection between two networks with different network part of the network addresses.
Many times there is also another IP address for management purposes, so you can say that it is
the router's own address, but it has nothing to do with the function of router as its merely a
convenient access.
So a minimum is one incoming interface and one outgoing interface each of which has to have a

2. different IP address to differentiate them. There can be as much as you want, taking care of many
network interconnections, provided you have the needed resources (RAM, CPU, network
interfaces, ...).