Math 110 Exam 3 All Sections March 713, 2013 No books, notes, or calculators allowed. Do NOT write on this exam. There is no time limit. 1. Given ! ! = 3! + 4, find the inverse of the function !. a. ! !! ! = ! ! ! − ! ! b. ! !! ! = ! ! ! + ! ! c. ! !! ! = ! ! ! − ! ! d. ! !! ! = − ! ! ! + ! ! e. ! !! ! = − ! ! ! − ! ! f. ! !! ! = − ! ! ! + ! ! g. ! !! ! = ! ! ! + ! ! h. ! !! ! = − ! ! ! + ! ! 2. Given the domain of all real numbers (unless otherwise stated) which of the following functions are onetoone? ! ! = ! ! ! ! = ! ! + 2 ℎ ! = !"# ! !, ! > 0 ! ! = 2! − 7 a. !(!), !(!), and ℎ(!) b. !(!), !(!), and !(!) c. !(!), ℎ(!), and !(!) d. !(!), ℎ(!), and !(!) e. !(!) and ℎ(!) f. ! ! and ℎ(!) g. ! ! and !(!) h. !(!) and !(!) 3. Solve for ! given the equation 3 !!!! = 9. a. ! ! b. 0 c. ! ! d. − ! ! e. − ! ! f. 2 g. 4 h. −4
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Math 110 Exam 3 All Sections March 7-‐13, 2013
No books, notes, or calculators allowed. Do NOT write on this exam. There is no time limit.
1. Given ! ! = 3! + 4, find the inverse of the function !.
a. !!! ! = !!! − !
!
b. !!! ! = !
!! + !
!
c. !!! ! = !
!! − !
!
d. !!! ! = − !
!! + !
!
e. !!! ! = − !!! − !
!
f. !!! ! = − !
!! + !
!
g. !!! ! = !
!! + !
!
h. !!! ! = − !
!! + !
!
2. Given the domain of all real numbers (unless otherwise stated) which of the following functions are one-‐to-‐one? ! ! = !! ! ! = !! + 2
ℎ ! = !"#!!, ! > 0 ! ! = 2! − 7
a. !(!), !(!), and ℎ(!) b. !(!), !(!), and !(!) c. !(!), ℎ(!), and !(!) d. !(!), ℎ(!), and !(!)
e. !(!) and ℎ(!) f. ! ! and ℎ(!) g. ! ! and !(!) h. !(!) and !(!)
3. Solve for ! given the equation 3!!!! = 9. a. !
!
b. 0 c. !
!
d. − !!
e. − !!
f. 2 g. 4 h. −4
4. How much money will you have if you invest $500 for 3 years at 8% interest compounded quarterly? a. 500! .!" b. 500 1.02 !" c. 500 1.08 !" d. 500 1.06 !"
e. 500 ! !" f. 500 1.06 !" g. 500 1.02 !" h. 500 1.08 !"
5. Given ! ! = ! + 1 and ! ! = !! + 3, find the composite function !(! ! ). a. ! ! ! = !! + 4 b. ! ! ! = ! + 4 c. ! ! ! = ( ! + 1 )(!! + 3) d. ! ! ! = !! + 2 e. ! ! ! = !! − 4 f. ! ! ! = ! + 2 g. ! ! ! = !! + ! + 4 h. ! ! ! = ( ! + 1 )(! + 3)
6. Solve for ! given the equation log! ! − 1 = 2. a. 1 b. 12 c. 5
d. 2 e. 10 f. 7
g. 0 h. 4
7. Which of the following equal(s) −2 log !!? a. log !
!!
b. − log !! c. −5 log ! + log !! d. – log !! − log !
e. both a. and b. f. both a. and d. g. both b. and c. h. a., b., and d.
8. How long will it take an investment to double in value if it earns 3% compounded continuously? a. ln 3 b. 2 ln(.03) c. !" !
.!"
d. −ln 2
e. −2ln 3 f. !".!"
!
g. !" !!
h. !! ln 3
9. Find the focus and directrix of the parabola ! − 2 ! = 8 ! + 1 . a. focus −1,2 and directrix ! = 4 b. focus 1,2 and directrix ! = −3 c. focus 1,−2 and directrix ! = −3 d. focus −1,2 and directrix ! = 1 e. focus −1,−2 and directrix ! = −3 f. focus 1,2 and directrix ! = 1 g. focus −1,2 and directrix ! = −3 h. focus 1,2 and directrix ! = −1
10. Match each function with its graph. I. II.
III.
! ! = log! ! ! ! = log!(−!) ℎ ! = − log! ! a. ! ! is graph I., !(!) is graph II., and ℎ(!) is graph III. b. ! ! is graph I., !(!) is graph III., and ℎ(!) is graph II. c. ! ! is graph II., !(!) is graph I., and ℎ(!) is graph III. d. ! ! is graph II., !(!) is graph III., and ℎ(!) is graph I. e. ! ! is graph III., !(!) is graph I., and ℎ(!) is graph II. f. ! ! is graph III., !(!) is graph II., and ℎ(!) is graph I.
11. What is the domain of ! ! = log!!!!!!!!
? a. ! ! ≠ 2 b. ! ! ≠ −2, 2 c. ! ! > −2, ! ≠ 2 d. ! ! > 0, ! ≠ 2
e. ! ! > −2 f. ! ! > 2 g. ! ! > 0 h. all real numbers
12. Find the asymptote and the x-‐and/or y-‐intercept(s) of ! ! = −3!!! + 1? a. asymptote at ! = 0; y-‐intercept at 9 b. asymptote at ! = 0; y-‐intercept at −9 c. asymptote at ! = 1; y-‐intercept at −,8 d. asymptote at ! = 0; y-‐intercept at 1 e. asymptote at ! = 1; x-‐intercept at 2 f. asymptote at ! = 1; y-‐intercept at 8 and x-‐intercept at−2 g. asymptote at ! = 0; y-‐intercept at −8 and x-‐intercept at −2 h. asymptote at ! = 1; y-‐intercept at −8 and x-‐intercept at −2
13. Find the equation of the parabola with vertex at (2,−3) and the focus at 2,−5 . a. ! + 2 = 8 ! − 3 ! b. ! − 2 = −4 ! − 3 ! c. ! − 2 ! = −8 ! + 3 d. ! + 2 = 4 ! − 3 !
e. ! − 2 = −8 ! + 3 ! f. ! + 2 ! = −4 ! − 3 g. ! + 2 ! = −8 ! + 3 h. ! − 2 ! = 8 ! + 3
14. Express 2 log ! + 1 − log ! + 3 as a single logarithm. a. log !!! !
!!!
b. log ! + 1 ! ! + 3 c. !"# !!! !
!"# !!!
d. 2log !!!!!!
e. log !!!!!! !
f. log ! !!!!!!
g. log ! + 1 ! − ! + 3 h. log 2 ! + 1 ! + 3
15. Find the solution(s) of the equation log!!+ log! ! − 6 − 4 = 0 a. −2 b. 8 c. 2 and −8 d. 0 and 6
e. −2 and 8 f. −8 g. 2 and −6 h. 2 and 8
16. Given ! ! = !! and ! ! = !
!!!, find ! ! 6 .
a. 3 b. 1 c. − !
!
d. − !!
e. 2 f. !
!
g. −1 h. !
!
17. Find the vertex of the parabola !! − 4! + 4! + 4 = 0. a. 2,0 b. −4,2 c. !
!, 2
d. −4,4
e. !!, 0
f. 0,2 g. − !
!, 2
h. −4,0 18. A rare strain of bacteria grows according to the law of uninhibited growth. Initially, there are 100 bacteria. After 4 hours, the bacteria count is 1300. What is the growth rate of the bacteria (given in bacteria/hour)? a. 4!"13 b. !
!
c. ln 13 − !"4 d. !"!"
!"!
e. !"!"!
f. !! g. !
!!
h. none of these 19. Use the properties of logarithms to find the exact value of log! 6 ∙ log! 4 + !"
!!.
a. 2-‐! b. !
!
c. −1 d. 0
e. 2 f. 10+!
!
g. 3 h. 1
20. Solve for ! given the equation 2!! − 2! − 12 = 0. a. 1 and 4 b. −3 and 4 c. 2 and −3 d. 1 and !" !
!" !
e. 2 and !" !!" !
f. 4 only g. 2 only h. no solution
Answers 1. c 2. d 3. d 4. g 5. a 6. e 7. f 8. c 9. b 10. f 11. c 12. h 13. c 14. a 15. b 16. g 17. f 18. e 19. h 20. g