NIAC 1147 Exam #3b
Name:
{)mbWtr ~t1
HONOR CODE: On my honor , I have neither given nor received any aid on this
examination.
Signature: _ _ _ _ _ _ _ _ _ _ _ __ _ _ __
Instructions: Do all scratch work on the t est itself. Nlake sure your final ansvvers
are clearly labelled. Be sure to simplify all answers whenever possible. SHO\i\T ALL
WORK ON THIS EXAM IN ORDER TO RECEIVE FULL CREDIT'1i
No.
1
2
3
4
5
6
7
8
9
10
Bonus
I Total I
Score
/8
/4
/ 18
/10
/10
/4
/1 0
/8
/ 14
/ 14
/ 10
/ 100 I
(1)
(a) For the given functions f and g, find the composite functi on (f
points) f( x) = x 2
+ 4;
g(x) = x 2
@x+ 10x +
4
(d)
X4
+ 21
2
g)(x). (4
+5
(c)
29
0
X4
+ 29
(e) None of the above
-:.. jW2-r~
-=:
('k2..~)' -r~
":: X~ -\-to ;'" 15 +~
-::;.. . X.'1 +(0)(;'1+-2-1
(b) Find the domain of the composi te fun ction (f
f(x ) =
(a) (3 , 7) U (7,00)
~7;
x­
g(.1;)
=
0
vx=3
(b) [3,7) U (7,00)
(d) [3,7)U(7 ,52)U(52,00)
-\~IJ.t
g) (x). (4 points)
®
[3 , 52) U (52,00)
(e) None of the above
,.-.
of ~ (() ~
x-e :f '{o,
-"'"
±l-
X ':fSL
x- ~"*O
ti:ll
X~1­
1)~r- J- ~(j{~~ : ~&.)=Fl
~*l­
(~ ~ -3") {;1"t 1)0""';'''
ot- (~.j' (kr [~, S"~ \J (S-J., <>0)
(2) Det ermine whether
0 1'
not each function is one-to-one. (2 points each )
(i) {(8, -2) , (2 , - 8), (3, 5), (-3, -5)}
e
Yes
(b) No
No
iif~ 1-~) (ii)
10
y
- 10
1)z,e-~
\~'1'L ~5t
5
-5
10
-5
-1 0
(a) Yes
N)-t
~
No
p~ S ~ ~+CJ>
(3)
(i) Given t he function f, determine its inverse. (4 points)
{ (- 3, 4), (- 1, 5), (0, 2) , (2, 4), (5 , 7) }
(a) {( -3 , -4), (-1 , -5), (0, - 2) , (2, - 4), (5, -7)}
{(4, - 3), (5 , - 1), (2, 0), (4 ,2) , (7, 5)} (c) {(3, - 4) , (1 , - 5) , (0, - 2) , (- 2, - 4), (- 5, - 7)} Cd) { (3,4),(1 , 5), (0 , 2),( - 2, 4),(- 5, 7)} ®
(ii) Given the graph of f , determine what the graph of f- 1 looks like. (4 points)
/
/
/
(a)
(b)
· 10
.j
- III
1!)
.j
10
I .,
I
I
1.10
@)
(d)
'°1
,
I
,'(
1
,
.'
I
·10
.,
.,
,
I
I
, ·5
1·5
!.,J
I
I
,
I
· 10
I
,
10
I
I
I
, . 1<,
10
(iii) Find the inverse of f· State the domain and range of f and f-1. (10 points)
f( x) =
~
x +2
jt_~
~)C
~=
)<t-l
Xt - 2
elf),) x =
I::
~~
~
2.)(
~
.f-'1 ex\= -~~x-
~ .f­ f ~ /
*- 3
~
7~ - "2.
c+ f-'
~
@ ffiw,bt' cJo,.Jt
(4) Graph the function. (10 points)
t-e 1L @\f\
( - (Alis
x
f(x)
=
10
@\~l
, e:f.·..2. ~
f,
(
t
I
/
--.-
-=-:=~
-.....::;:....,,;::;;;...~- -
------.---­
-10
5
10
(5) (i) Change the exponential expression to an equivalent expression involving a
logarithm. (2 points)
(a) loge; 3 = x
(d) log3 x
=
~ one of the ab ove
6
(ii) Change the logarithmic expression to an equivalent expression involving an
exponent. (2 points)
~
In x = 7 ~ ::.> l~e IC ~ "+
e :::: X
(a) x 7
=e
(b) e:
t
=
7
fYe x
7
=
(d) 7 e
(iii) Find the exact value of the logarithmic expression. (6 points)
1
log16 - =. 'f..
8
\
~
=
x
(6) Find the domain of the function. (4 points)
f(x)
(a) (-64,64)
(d) (-oo,-8)U(8 , oo)
6~_ '(L >0
(~n')(~-'x) >0
JtK:::-o
-
:1.
~-~-~
f\\v\t ~(
~:ro
-r}t
........­
'X:=-~
""-V\+ / ~
Yrt1. '(~O\· (~)\ F~O>
Lr»b j
fn,-', ~oill~
l)o~"
: (- ~ 1'6)
®
=
(-8,8)
log3 (64 - x 2 )
(c) [-8,81
(e) None of the above
@ nHe~\)J\ ~\ IC"' CVtl~
1
(7) Graph t he function. (10 points)
f(x)
=
-log4 (x - 1)
J
+3
4ve~
G? (i,~-t- I
10
!t
(1
I '
5
-10
-5
10
~ - \O'\f ~-1~-t)
-5
-10
~
'- '­ -® -lOj't (~-l)
(8) (i) Write the single logarithmic expressions as a sum and/or difference of loga­
rithms. Express powers as factors. (4 points)
(8x3)~1
logs
+ 3x
(x _ 6)7
,x > 6
(a) 5 logs x + glogs (1 + 3x) - 710gs (x - 6)
(b) 1 + 3 logs x - 5 logs (1 + 3x) - 71og s (x - 6)
~ logs 5 + logs x 3 + logs (1 + 3x) -logs (x - 6) - logs 7
t
I~% Ct\<~) ( It~l + 31~:X ~:_1:~~1 + 71og ~+6~\01! ~ IOj ~ (It:s.)- +I~& (X-I,,)
8
3x) -
(x
l(.-
IQjs [~.') + IOj ~ (It 3)<) v, -Io~h (x-ro)'+
IlIj %~ + IO:hX'l t \0'\ (It-1><)* - bJt (); _1,)1
,
L-Y-
J%
(ii) Express as a single logarithm in simplest terms. (4 points) 36 logg Vx + logg (36x 6 )
(a) logg
x't (c) loggx¥ -
logg 36 (b) logg x¥
&
loggx 10
!~,(~)
(9)
(i) Solve the equation. (4 points)
5x - 4 = 2
(a) 4+log 2 5
(b) - 4+log25
~ 4 +l0g52
(ii) Solve the equation. (10 points)
32x
+ 3x +1 -
4
=
0
~
3Y..-3' :: ~ · 3X
u.1.. t3u - Y. -::- 0
(v.~ ~I(\A- \) -=- 0
U;:[O
:j..
\A-=- -~
3'K -=--\.{
X -=- l.j~
(T
f\cfr~
u~t~
U ':- \
~X -:; \
X~lOj3
G
'
(d) -4+log52
(10) (i) Solve the equation. (4 points)
6 + 41nx
(a) In (1)
(b)
lV\
41~1
k
7:
= 15
@ e~
~ <E-~ lDJ~ X ~ ~
<1 '>
~
e-rt :; )c
(ii) Solve the equation. (10 points)
log5 (x
lv~'5 (lt4--3) :. l - II9~S 't-t)
+-\GjSeX- !')
t 101, ~-\)
100t ( X~3) 4--JS
·
\oa. (x-t\ ~ l
JS
\o~ <; ( ('(f-3)(l\-~\ ~ I C" t'"~} (x-\) ~ 5' '1
'I..
~2X - <\ :; O t~ -I'l{) (~ - J) :::. 0 + 3) =
1 - log5 (x - 1)
(c1)
~
Bonus. Solve the equation. (10 points)
logx2 + 3x+2
""
,,- 1."l
x. - 2'1.'J.. -\'\ h - 1 -==
+11 '( - 2
_x2.. -3x -2..
'("- 2)1;3 - 2)(2
----.-.
==
(
(X4 -
2x 3
-+1
2X2
+ ll x
- 2) = 1
~'. X::\/ ~ttor:
'l-\- !)~t:l) I
~'2.
­
t­ Jl t
t
- ;)
~
\
,
- - -.:i. -~ ­
X~-2X~-'"S)(-a.-\-%'x -4-=-0
-(
- 'i
-'t
~
~(x)~ ()c-t)(X~ -
<i,
Xl. -
,
-f(x)
--.r-
-'-\ Y
lliJ
4x+4)
~
5~)
j (--I) -=: (-I\~ - db\ ~(--1)'L {- &h )-'1
1
~-=-4
f05\-\\'vt.
-~
--I
x-I
=3
::. \ -;Z (-I)
\Y \
-3(,) -\-S(-I)-~
:::- I +- ~ - ~ -~ - ~ "4=-0
+C-X) =(-x)" - :1(_'1.)'3_ 3C-x)"z' t-~ (-x)-~
=-X'l+-;l.)(3_
-
~(-J)::: (_~)'i -;;'(-J.)~-~(-;}.y t i (-:2)-'1
3)(1..-&x -"\
=1~-.;2(-f)-3(4) of ~(-2)-~
-:: (\0 +(~- \1.. - (lc-~
~~ of- ~ _
l,2., ~_
\
'l..
-\ ' -
\
f0S5~~\i~~
\..I
...l- -::
(t
::: {' ~21 ± Uc
I
-rG) ::- J~ - d..(\)~ - 3(\)2 +-~l\)-\(
~\-L-~tY.-'1
I
t, 2,
\J
I
-=--D
~'. x.~ - ~/ ~G..fo(-: 'X-(-;t) -=X+-;l
-~\
- \{
~ -1.
(p
.-~ -~
2
\
-t
~
I
lQf
f(x):: (X-I)( )(f~)(~.l_ ~l\:+l) ,
~
(~-;>.) (}(-I)
)(-1 =0
itt!X =:'2.
).,
)c-
V=-O
Jt .tL
x=1
Scratch Paper