Chapter 11: Rational Expressions and Functions
Sectionll.l
1. A rational expression is a fraction in which both
the numerator and denominator are polynomials.
x+3
Answersmay vary: x 2 -2x+5
3. a. Substituting -2 for x in the expression will
give a denominator of 0 and division by 0 is
undefined.
15. ~+
x2-64
=~*x2-4x+4
5x-10
x2-4x+4
5x-1O
x+8
(x-2)(x-2)
=-*
5(x-2)
_~~(x-2)
- 5~(x-8)~
b. Yl = (X - 3)/(X + 2)
-- x-2
c. When X = -2, the calculator reads ERROR for
Y1.
3
4
3*5
4 *5
5. a. -=-=-
(x-8)(x+8)
5(x - 8)
Skills and Review 11.1
15
20
17.
f(g(x))
= 1010&(%)
= X
g(f(x)) = log(lO%)= X
19. a. log(x) =4
1010&(%)= 104
c
.
7. a.
x-5
(x-5)*(x-3)
x+3 - (x+3)*(x-3)
-
2 1 2*4
1*9
-+-=-+-=-+-=9 4 9 *4 4 * 9
8
36
x2 -8x+15
X = 104 = 10,000
x2-9
9
36
17
36
b. log(x)= -3
1010&(%)= 10-3
x =10-3 =- I =., 1 =.001
10-3
4
x+l
l*(x+l)
(x+2)*(x+l)
x+l
1
c. -log(x)
5
1
c. -+x+2
log(x)
1000
=1
=5
1010&(%) = 105
+
+
(x+2)(x+l)
5x+9
4*(x+2)
(x+l)*(x+2)
4x+8
(x+2)(x+l)
(x+2)(x+l)
X = 105= 100,000
21. a. The number of eagle pairs is increasing. The
base (1.065) of the exponential function is
greater than 1.
b. Because x represents the number of years
since 1994, the output for 1994 is the initial
value, 4449. There were 4449 eagle pairs in
1994.
6
3*)
3
9. a. 32 = 16* ) = 16
c. 1+ r = 1.065
r = .065 = 6.5%
The number of eagle pairs is increasing by
6.5% per year.
c x2 +3x+2 - (x+2)~
.
x2+x
- x+2
x~-~
11. Method #1: Compare table outputs. Enter the
expressions in the Y= menu. The table outputs
for Yl and Y2 should be equal except at X = 5,
where the fIrst expression is undefIned.
Method #2: Compare graphs. The graph ofYI
should be the same as the graph ofY2, except at
the point where X = 5.
d. In the year 2008,x = 2008- 1994= 14
y = 4449 *1.06514
y = 4449 *2.4149", 10,744
If the number of eagle pairs continues to grow
by 6.5% every year, then there will be about
10,744 eagle pairs in the year 2008.
@ Houghton MifflinCompany. All rights reserved.
-
- ---
23. a. The graph of the basic absolute value function
f(x) = Ixl is shiftedto the left 1 unit.
b. The graph of the basic cubic function
f(x) =x3 is reflected across the x-axis.
c. The graph of the basic square root function
f(x) =.[; is stretched away from the x-axis.
d. The graph of the basic reciprocal function
f(x) =.!. is shifted up 5 units.
x
25. a. Substitutex = 0 into f(x) = x2 - 3x - 8. The
y-intercept is (0, -8).
b. Solve the equation 0
= x2 - 3x -
8. Using the
quadratic fonnula, a = I, b = -3 and c = - 8.
x=-:tT3)
2*1
~CW -4*1 *(8)
2*1
x = ~:t .J9 + 32
2
2
x=~:t J4i
2
2
.
3 J4i
The x-Intercepts are ( "2+ 2'
0) ,
~_J4i
(2
2'
0
)
.
-b -C3)
c. x=-=-=-
2a
3
2 *1
2
The vertex is
~ - ~ = 1.!.
( 2'
4
)
(
2'
)=
-1O.!.
4
( 1.5 -10.25)
'