Name:
A;NsIAlE e.. t<Ey
Class: _ _ _ _ _ _ Date: _ __ ID: A
Unit 7 Review - Radicals and Rational Expressions
-Cl I.
~
,....-t
l'3Ji) 'f
SimP;iIY 8'".
a. 2
b. 8
~
c. 2 'i ~f!t.p
~
32
3
16
True or False: .
2. For any integer a
* 0, a
II
n
1
an
=-.
=if;;.
3. For any integer n > 0 and any positive real number a, all n
TrvL
6. 2V 128 + 3V 1024
8.
V6x y V4x s
3
7
•
9. The area of a circular park is 2 x 104 square feet. The radius r (in feet) can be approximated by the model
r=
(A)
1/2
where
Ais the area ofthe park (in square feet). Approximate the radius ofthe park. Use 3.14 for
1l n. Round your answer to the ne~~st whole number.
r::- (2)()OLfT)\}i~ 1'1,1 ~
rtl)~ 2(2)-:~
3
10. Let r(x) = 2x and s(x) =x -3. Find s(r(2)).
~Df
S(~)::: tt ~ -3:;
i·
lID
11. A discount function D(x) that takes 10% off an entire purchase can be given by D(x) = 0.90x where x is the
amount of the entire purchase. A tax function 1{x) that adds a tax of 10% to an entire purchase can be given
by T(x) = 1. lOx. Explain what D(T(x» and 1{D(x)) represent. Then compare the two compositions.
fLfttP1rS Tcq(f(l CtV\ ,i~/ «rhtv1 tak~ " lo'O~ Glt$(clM-1
V(\ tC> ~ ) -= 'I 0 til I0 )<) ~ I <J~ ;X
,
pL11xl)
I
i
"
\ (f)(X)) (tfn:)c~~ ~
\ C,~D,x) -::
..L I, '
I q",~
<
'"
loot dIS((lV"l-t, f~ to.xt,(] d-k \~
1,)0 (/1D ~) -;. ~ qqf.
I
_
I
r
1
•
t \.t c.)l'v\po)d it"'~
tl'tL
Cf wi.
I
ID: A
Name: -------Let f(x) :::: (x - 3) 2 andg(x) =
-t( ~~) :: (-~- 3) '2.
12. Jtg(x»
13. g(f(x»
~.
Perform the indicated operation and state the domain.
x
't(fi<-3)'l.)::
~(-~)
14.g(g(x»
a
-:.. . .
~/'l-::~
V'.xf=O
p~ x/;3
~-1'_;':: X P'IXto
(
~
~
15. Find the inverse ofthe relation (-5, 8), (8, -5), (-6, 1).
.!rwtr5(', ',Jit~;x~ j .
16. Write an equation for the inverse ofthe relation y:::: - Ilx + 9.
=_x 2 -1. Is the inverse ofj(x) a function?
17. Sketch the graph of the function f(x)
~\(..I'S(
_1-+---7
-1 ... 2.
...;z "I I)
-2... I
(j -,
-2
Open-ended:
18. Explain how the inverse of a function is determined algebraically. Give an example to explain the steps you
SWltch X~
follow.
'f) 14Vl SolVL~r y' LoD~ ~t-#-I(p,
Graph the function. Then state the domain and range.
19. f(x)
-3Vx-1 -3 X
D(~!t~/>. L~~l,<;:r\f-'s)
Q~~f~5
J
r" =3
(xv,~1o~"Ih x~{.~'f~ ,~)'f "if q;gSZ,J
21. Which gives the solution of
...,.
3
~) -~ h.
--7
V5x + 4 + 5 = 6?
{:§X~J~ t 3
{\
bXf t.f ::
t
-5
I
-i
A;: - 3
c.
d. none ofthese
-~
5"
y~ ··3/~
I,.
1
2
3
0
Co
2­
C~k',
rl't
3 (1S~
3 ' (p't
.
-"5-5
-7
I -3
Solve the equation. Check for extraneous solutions.
20. \
!!-
(~2
~ t~ 2 i f
-G,
Name: __________________
ID: A
d~tI q
22. Solvel~ 5x - 4)= 2.
23. Solve
SA:N~ .fY;-~
)
;?~~ ~tE.r.f)~
((f)1j ~§, ~;:2~I(fi
071"tV -J 2x + 1 ::;:: ~ + 2. Check for extraneous solutions. sa.. a:#4(J~
24. For a mammal, the heart rate r (in beats per minute) and brain weight w (in kilograms) are related to theoooy
mass m (in kilograms) by the functions r ::: 241m-1/4 and w ::: 9. 1m3 / 4 •
~,
a. Solve the fimction r = 241m-II' for m.
b.Solvethefunctionw==9.1m
3/4
form
&;J;~ -JIij -¥ ~..,"; &~l~
1_~\~(Wl3/'f)1'~
LC,·V '-
/
-:?!
-
vi) ~3
I
l
;::
\
€
c. Explain how to use the answers from parts (a) and (b) to express the heart rate r as a function of the brain
4,
d,
weightw. Then find the function.
S(t et:;~ti.:>"l)
hi)
{YOM~.r-ts ·o.~b)
J -rtw'lselvt...
~r
.__ ._ -~.-'~---'.. ~-
y"" I
25. It takes a train 8 hours to travel from Capital City to Johnson City when it travels at a speed of 65 milh. How
long would it take the train to go the same distance when it travels 40 milh?
CV
13 h
c.
b.
11 h d.
183 h
8
13 h
J')l'-'S: 5z..o X -I.£D ~5 ("Z) X-:-I'3. i,
26. The wattage rating W (in watts) of an appliance varies jointly with the square of the current I (in amperes) and the resistance R (in ohms). If the wattage is 6 watts when the current is 0.2 ampere and the resistance is 150 ohms, find the wattage when the current is 0.3 ampere and the resistance is 300 ohms. a. 180 watts c. 27,000 watts
W _ 1<. 2.
W= 1'3 2 • 3DO
b. 90 watts
~ 27 watts
,,,\
~ 2'2.'/51.,.1 W"::.u't, 3tH)
27. Write an equation for the relationship: ~ "" fb k ~ k. =7
y varies directly with the square root of x and inversely with t y -:z.. 1<- fi. 28. The intensity I (in f~-candles) of light received from a source varies inversely with the square ofthe
distance d from the source. If the light intensity is 5 foot-candles from 16 feet, find the light intensity from 19
feet. Round your answer to the nearest hundredth, if necessary.
5-=..h ,;;j ~: IlYO ! ::.. (J-BO
:n
1(..2. 1'12,.
fix 7
(;I"
-:t~3 ;OS­
29. Let f(x) ::;:: 2x + 1 . Find the asymptotes ofthe graph ofI, and tell how the graph is related to a hyperbola with
:c f<.
,=- ..
I
w";. z..
I:. /(
.
+
a
equatIon of the form y :::: - .
.
111\: st1 kr, ....\;I\£du ~ 0
.,,,-
2X~~::
)(=
Identify the vertical asymptote(s) oftlle grap
the function. X
d
f
vA,
30. f(x)
a. x = -4, x
b. none
Ji = -2,
x::;::-1
'I
3
.,
1..
(J,
'J - 2:
S<..t o'.L1-Ul,,".P.v:tor
""'2k! X4f:,· -l-f
x =-4, x ::;:: -1
~ x =-4, x=-2
c.
I.
HIJ ~ C«.AftL
i~ V'N~ ..",,1or ~ k~r4t ;~
..J
" •+J ,,~~
~ ~
~:: ,-,
__-.,...... "U...'1\Wr\.I\I\..'·'Otl"
-vI"
:::3'J
_..,L, ,-::..1
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
9xl.---.-J
z-3~x 'I"'~)(-I ~ ~ 0
,--_J
9xlx- tt) +-1 [)<:'f )~()
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
Name: _ _ _ _ _ _ __
ID: A
2
x
31. f(x) = x 2 -4
Identify all vertical and borizontal asymptote(s) oftbe grapb oftbe function.
32. f(x)
_ _)(3
_x 3
=- 3 ­
X
'1~ :;~ ,,-I
HA,
-{1\-2)l X--L+ Zx +'-I)
-8
cN.d: (otfht"C!t\.-t
~~"~f.-"".\'_'~'~~""'~'~''''
lY=2f
2
.
I fun' f() 3x 4x + 9
33 . Use t be ratlona
ctlon x = 5x2 + &x _ 3 .
'(":l-~
a. What is the horizontal asymptote off(x)?
a. Give an example of another rational function with a quadratic polynomial in the numerator that bas the
same horizontal asymptote as f(x) .
3
~x~
S~ 't~l~ tv,
b. Write a rational function with a cubic polynomial fn'lhe denfmin\tor that has the same horizontal
3(x)::-
asymptote as f(x).
hlx):: p X ~
Q,X
x2 +4
34. Compare the graphs of y =
behavior in your analysis.
4
x -16
and y
x: -x-12
=
~
2
x +4
.)(,2-'-4
...
x~ - 4.r + 4
Include x- and y-intercepts, asymptotes, and end
kJo
.....
~~~o
_<)-if.+32J
~2)b<.-t.)...L
,
X-'~T
\r~t', (.0)
vA',
I\A!y~O
: t\
lJo vA
frf'fi ~,ks
1)$~P1l,., Tx _ ~.l
ltlf'W r-td hl
}5'x- k r - t ...3
Divide the expressions~Simplify the result. -­
5x
-.hI 37.
0)b.
x-6
-3
x-
y~f rt1 t, (O) -J.1)
z, x----,
-el'l.cl b.-"'ctlJ; j r
x- 2 -::(~. P.~
x~
,i/v .,\' , ")
~ ~ l){r:tH/J~~i
X"'i>d', (2)0)(-2) 0)
..!i)
.,~
:'(OJ .~) " .
('X+S)(»?r) - I lX1S)}
I S x - ,S·)<·
r-?
•
)(Zt-'-f
"""~~~",~",-",,,~
36.
>Nt.$.)
~~,t I\IiI'\Ua~ ... ~ 0
X4 -16
'r- lx'a~)&tt~J ­ tJ<-2.)l)<~1)V<"t4)
(~lX~32
x +x 20
(rw.d
L~ j~. Co!lfi'(i (",,"'t. S )
Simplify tbe rational expression, if possible.
35.
X; ;
c.
x 9
--3
x-
Perform the indicated operations. Simplify the result.
4
d.
x+3
x-6
,-n.et
a;;
l OJ -~J
/tvo tiff
~l-v...vioV"' ".
Uf·~~vr'
(leAlc.s lilctL
~rt..~t)~~
)
Name: ID: A
-----------------
. 2x2-11x+12
. a1
39. Use the ratIOn expression 2
x -7x+ 12 .
~2x~.:~~0~~~~ ,,~1-~(](...t()_lQ3)~~
...~.
(X--4)(X-»
bt'tt} (X--3)
~j[ '-3'
a. Factor the expression. Then simplify, if possible. _
2x3+ x 2-76x+l05 3
2
? Explain.
c- X -x -41x+ 105
x..-::: -7, '{?J; X,,;};
13<>T-t~"" '.
b. Is the expression in part (a) equal to __
\
to'f \
uyaS /Art..
•
•
.
..
•
L.
•
:1:t~ll!.t~.1.
r:r--ll
-(.5
i
2<.~
V _,~)/
(2x-3/~T1 I~:-"""r'\
-.
\.1':::7_.;).,}
(~()(-?'J) ~
X:.y
<
2
~j
~..lS...-.
X-3
, c. Descnbe the error 10 snnphfy10g the ratIOnal expresslo~j.~~~1,;",:;::.._>
Y<-S k():.\.l,..s<().JA.Lf
2x 3 2x -3
)lX-)lf2 ~
2
\'~'---"'''''''---'''''--''''IC
~
v~
'U,."f..""S
."""'""".,..",,'''
.Wn\i -=-3
=-+_3=2+1=3.
'/{..-1!:-- -;:.--lS + t} _v(1\f-3)-::.1,r/\,,,:~)r
-);:",~!::::':::".,-.".~,j
x
x\.3
- 1\ J\.
,;-/'.\.} , .
----.
\;>.1 Lc9
.
"xx: J ~'/,;,_7~. -:::. 2X,z-,"x ". XL". %~ ) x,2_1.4.;:. C2.-:-.j)(";
Cj
I
'lfI.t
~2
~l,J< -Lf.(Z
t1£ XlX-(p.>":-Q';;f )('::.(.J
40. A company is designing packaging in the shape of a right cylindt r for a new product. They want the raoius of
.
\ fr; (/~ .. 7.. \ ~
the cylinder to be 2 inches less than its height.
l':;.. h - L
,
v" il ~ r, a. Find the volume of the cylinder in tenns of the height.. V-; ~ ( ~-'l)'l. ~
:5f\:'1f11"'2-\- b. Find the surface area of the cylinderin tenns of the height. 5.~\ "- 2f1 ·(h':t)"l.~"2 ( 'Z) k
21iv-h c. Write the ratio ofthe volume of the cylinder to the surface area ofthe cylinder in simplest fonn.
d. The company decides to choose between a height of 8 inches or 10 inches. They want to choose the one
which will be the more cost efficient. Which should they choose? Explain.
!C,~ ~
Perform the indicated operation(s) and simplify.
5A -
': )(*5'
(X75)~;) ~~
5x
2
2x
27x-18
c.
3x
b. d.
2x+5 2x-27
5x
2x
5x-27
Solve the equation. Check for extraneous solutions.
c, 44.
a
a-12
a.
a
4
+0'
1
b.
1
,
7 .J,...
Q 'Z-
...
l,i.
-;;:to
-:­
" )\ Q
,
(33 4
d.
\
Ca.-fZ )
""-""\,
+- ~ (C\ -l '2 ) ~ oJo,- \ )
+L\ at. - ~ S ,. c.\J~. IZ "'~.Q'~'
5
a=3
2
Name: ID: A
------------------
45,{~5+ 5
\e+
h,
a,
b.
0
x-y= 1
16
3
b.
G~ t'L ~ If
pr-tdvet ::c.
c.
d.
y=-3x
c.
6
(~\ 12
10 48. The variable z varies jointly with the variables x and y. When z = 60, x
and y 4. Which equation relates x, y, and z? kx~
I
80x
d. z:::::­
y
49. What are the equations for the asymptotes of the function y = -2 - 5?
x+
_
_ \
c. x-3,y--5 \
d. x = -3, y::::: 5 \
50.
a. b. c. d. Z
\iI ': ,.
!
3
,\·.....8
'/ z=5xy
b. z=72-xy
{',
t
,
X'I'" 42:
/
L,
'"
Y
-=7
x
= 8 when x = 6, what is y when x = 4?
a. z =xy+48
0.
J
P - .,
t ~ ... '
•
CD xy=6
47. IfY varies inversely with x, and y
a.
L
-1
46. Which equation shows inverse variation between x and y?
a.
~
e-
domain: x -:1:--1; range: y > 2
domain: x -:I:- -1; range: y < 2
domain: x -:I:- 1; range: y > 2 ~
domain: x -:I:- 1; range: y > -2 \
r-~",y.<.
\:S f.
6
.,..
HA'
,,(:-5:
.
, vI,
,
.
Name: _________________
~l
ID: A
51. Which graph represents the function y
= x+
-1 + 4?
2
a.
"(1\\ y..--:­
ttEY \/-:: 4­
c.
I
l---­
L __
1­
t
Iy
-
j
I
~
-----
._­
~
--.
~-"--
--- t----­
1-· 1--­
----­ --.
1
,
-­
----. 1­
1-7 vI""
I
I
I
I
I
X
J------j--­
-.- t-­
d.
t
!
l'
L
J-'
j
:/
I
!-
-
~
.­
~l
F
1--­
f--­
-_.
i_
-T
I
i
l
:
I
\
y
i""""
-----­
f---­
-­
I
I
I
;
I,
:
il x
\'2.\)
3~
_ 1~.J.t,! f-_'
fe i' V \--1.1 -\
2
J;./ 52. Simplify the expression 3x +21x+36
'-"1
&+24
'"' i
d. 53.
a.
x+3
c.
(J
x+3
2
d.
2
x-5
MI'
x 9"
u tiP1Y: ---3­
2
x+
x +2x-15
a.
1
b.
(x 3)(x 5)
(x +3)(x+ 5)
(
'>
~.'J
(x-3)(x-4)
2(x+4)
1
2
.
.~
c.
@
7
!
L
x 3
x+3
x-5
x+5
'::
Name: ________________
ID: A
2
<i(X2+~-!)
LX -Z)(t:- L )
2
/)
..
9x + lSx-72 -""- 3x + l5x+ 12
1.11 54. DIvide: x2 -4x+4 . x 2 -3x+2
10
@:/
3(x 1)
x+ 1
b.
27(x+4)2(X+ 1) (X_2)2(X_1)
c.
-~fw~·!
; J{.,.i,:
x 2 -64 x+ 3 55. Simplify x+8
-.~
lli
x-I x+ d.
3
c.
x+8
tV
x-8
X+i
lk~""6
."~r .... _''-''
x+
a. b. (x + 8)2 (x - 8)
(x +3)2 (x- 8)(x -8)
x+3
2x+3 x+l
56. Subtract: x + 2 - x + 2
C,
a.
0
a.
12x+ 1 b.
12x+ 1
x 2 + lOx + 16 b.
x+2
f'
(\
~
'"'-.
6
.
--
6(x + 2)(x + 2)
r'
<•.;
>(+-1-t)~fj
•
':)
:.'.) 0
(9
7
c.
5
-':;>x,:;:
d.
14
2x
= x-3'
b. 3
,
i
to, 1-\
":
7x 2 +39x+2
6(x + 2)(x + 2) 5x 2 +52x+6 ~,. .. ­
-3 a.
J'
"
I~X.f't
;
b.
A.59. So]ve4+ x-3
X-s
,".-
-7 a.
c.
d.
(8 .:15 .
,,-.-----,'
n
U"':+.
58. Solve iX ~ .li:<~'~ +1\
3x+4
x+2
d.
'-v
!- x~ 0)
)(-.;' j
"
.
8
1
\,-~--, - ,
1
'10 ~..
1
l~ no solution
Name: - - - - - - ­
C\.
ID: A
60. Solve x __5_ _ -3
3 x + 4 - 3x + 12 .
(it, -6or2
c.
0:
d. 2
6or-2
-6
-
..~
<:~",::t._
.~
.J.~
® x~ 2
.l: _ ~
3.
(.
/
~
2-t.t{
2
./ 9