7.2 Reducing Rational Functions

624 Chapter 7 Rational Functions
Similarly, if we insert x = 4 in the left-hand side of equation (15),
2x − 6
x 2 − 7x + 12 = 2(4) − 6
(4) 2 − 7(4) + 12 = 2 0 .
Again, division by zero is undefined. The left-hand side of equation (15) is undefined
if x = 4, so the result in equation (15) is not valid if x = 4. Note that the right-hand
side of equation (15) is also undefined at x = 4.
However, the algebraic work we did above guarantees that the left-hand side of
equation (15) will be identical to the right-hand side of equation (15) for all other
values of x. For example, if we substitute x = 5 into the left-hand side of equation (15),
2x − 6
x 2 − 7x + 12 = 2(5) − 6
(5) 2 − 7(5) + 12 = 4 2 = 2.
On the other hand, if we substitute x = 5 into the right-hand side of equation (15),
2
x − 4 = 2
5 − 4 = 2.
Hence, both sides of equation (15) are identical when x = 5. In a similar manner, we
could check the validity of the identity in equation (15) for all other values of x.
You can use the graphing calculator to verify the identity in equation (15). Load
the left- and right-hand sides of equation (15) in Y= menu, as shown in Figure 1(a).
Press 2nd TBLSET and adjust settings as shown in Figure 1(b). Be sure that you
highlight AUTO for both independent and dependent variables and press ENTER on each
to make the selection permanent. In Figure 1(b), note that we’ve set TblStart = 0
and ∆Tbl = 1. Press 2nd TABLE to produce the tabular results shown in Figure 1(c).
(a) (b) (c)
Figure 1. Using the graphing calculator to check that the left- and right-hand sides of
equation (15) are identical.
Remember that we placed the left- and right-hand sides of equation (15) in Y1 and
Y2, respectively.
• In the tabular results of Figure 1(c), note the ERR (error) message in Y1 when
x = 3 and x = 4. This agrees with our findings above, where the left-hand side of
equation (15) was undefined because of the presence of zero in the denominator
when x = 3 or x = 4.
• In the tabular results of Figure 1(c), note that the value of Y1 and Y2 agree for all
other values of x.
Version: Fall 2007