STUDYGUIDEFORTESTI
MATH1113
1.
Determinewhichofthefollowingarefunctionsarefunctionsofx.Giveareasonto
supportyouranswer.Ifitisafunction,statethedomainandrange.(Assumealltables
arecomplete.)
(a) (b) x
y x
y
3
1 4
2
7
11 2
7
3
−3 7
−2
−1
5 −2
4
(c)
(e) 8
(d)
(f) 2.
Determinethedomainofthefollowingfunctions.
4
3
(b)
(a)
(c)
3.
1
(d)
√3
√
Determinewhetherthefollowingfunctionsareeven,odd,orneitherbyevaluating
f`(−x).Checkyouranswerbygraphingandlookingforsymmetry.
(a)
99
(b)
(c)
4
1 (d)
| |
4
5
(f)
3 (e)
3
(g)
√1
(h)
|
3|
2
4.
5.
6.
| | if
2
.
2 if
2
(a) Findf(−2),f(0),f(2),andf(3).
(b) Sketchthegraphoff.
Given
1 if
2
3 if
2
0.
3
if
0
(a) Findf(−2),f(−1),f(0),andf(2).
(b) Sketchthegraphoff.
Given
Determinetheintervals(x‐interval)inthedomainonwhichthefollowingfunctions
areincreasing,decreasing,and/orconstant.
(a) (b) (c) (d)
3
7.
Determinewhetherthegraphsofthefollowingequationsaresymmetricaboutthex‐
axis,they‐axis,theorigin,ornoneoftheabove.
(a)
4
(b)
4
(c)
3
5
(d)
1
1
(e)
8.
Let
2
andstatethedomain.
(a) f+g
(b) f−g
(c) fg
(d)
(f)
1
9.
3and
.Computethefollowing,simplifyifnecessary,
Computef◦gandg◦fforeachofthepairsoffunctionsbelow.Also,statethedomain
off◦gandg◦f.
(a)
1
3 ; (b)
36; (c)
; √
√ 10
10. Verifybycomputingf◦gandg◦fthat
and
areinverses.
11. Sketchthegraphofeachofthefollowingfunctionsanddeterminewhetherornot
eachfunctionhasaninverse.Ifithasaninverse,sketchthegraphoftheinverse.
(a)
(c)
1
(b)
√
5
4
12. Findtheinverseofthefollowingfunctions.Sketchthegraphoftheinversebyusing
thegraphoff(x).
(a)
(b)
√2
1
13. Computethedifferencequotientforthefollowingfunctions.Usethedifference
quotientformula
14.
(a)
(c)
If
(a)
(c)
(e)
(g)
(i)
2
,
0.
1
(b)
3
2 1
2,find
f(3)
f(−x)
f(2t)
f(2t+1)
Iff(x)=4,whatisx?
15. Thegraphof
following.
9
(b)
(d)
(f)
(h)
(j)
2
f(−5)
f(t)
f(t–3)
Ifx=8,whatisf(x)?
Iff(x)=10,whatisx?
1isshownbelow.Sketchthegraphofthe
(b)
(d)
(f)
(a)
4
(c)
(e)
3
2
5
16. Findthex‐valueswherethefollowingfunctionsarediscontinuous,identifyeachasa
verticalasymptoteormissingpoint,findthehorizontalasymptote(ifany).Findany
obliqueasymptote.Sketchagraphofthefunction.Plotandlabelthex‐andy‐
interceptsUsedashedlinestorepresentasymptotes.
(a)
(c)
(e)
(f)
(b)
(d)
Hint:Thenumeratorfactors.SeeSectionR.5,Example3,page44.
(g)
ANSWERS
1.
2.
3.
(a)
(c)
(e)
(a)
(c)
no
yes;D:(−∞,∞);R:(−∞,∞)
no
(b)
(d)
(f)
yes;D:{4,2,7,−2};
R:{2,7,−2,4}
no
yes;D:(−∞,∞);R:(−∞,∞)
D:(−∞,∞)
D:(−∞,−3)U(−3,9)U(9,∞)
(b) D:(−∞,3]
(d) D:(−∞,1)U(4,∞)
(b) neither
(d) even(symmetricwrtthey‐
4.
(a) even(symmetricwrtthey‐axis)
(c) odd(symmetricwrttheorigin)
axis)
(e) neither
(g) even(symmetricwrtthey‐axis)
(a) 2,0,2,2
(b) (f) odd(symmetricwrttheorigin)
(h) neither
6
5.
(a) 1,−2,−3,−3
(b) 6.
7.
(a)
(c)
increasing:(0,∞)
decreasing:(−∞,0)
increasing:(−∞,∞)
(b)
(d)
(a) x‐axis
(c) y‐axis
(e) origin
(a)
2
3
; :
∞, 3 ∪
3, ∞ (b)
2
3
; :
∞, 3 ∪
3, ∞ (c)
(d)
9.
(a)
(b)
∘
∘
(c)
∘
∘
10.
11.
Note:Theanswerisinworkingtheproblem.
1; :
∘
3
∞, 3 ∪
3
1
∘
(a) no
(c) no
increasing:(0,3)
decreasing:(−2,0)
constant:(3,∞)
increasing:(0,∞)
decreasing:(−2,0)
constant:(−∞,−2)
(b) y‐axis
(d) x‐axis
(f) x‐axis,y‐axis,andorigin
8.
2
5
; :
√
√
√
3
9; :
∞, 3 ∪
∞, 0 ∪ 0, ∞ ; D:
√
3, ∞ ∞,
∪ 1\3, ∞ 10
36; D: 0, ∞ 36
10; D: √36, ∞ ; D: 0,100 ∪ 100, ∞ ; D: 10, ∞ (b) yes
3, ∞ 7
12. (a)
13.
14.
(a) 2
(c) 2
(a)
(c)
(e)
(g)
(i)
(b)
(c) 6
3
2
1
4
(b) 28
(d)
2
(f)
7
10
(h) 54
(j) x=−3,x=4
2
2
2
4
4
2
2
x=−2,x=3
15 (a) shiftsrightby4anddownby3
(b) verticalcompressionbyfactorof 16.
(c) horizontalstretchbyafactorof3
(e) reflectionacrossthey‐axis
(d) reflectionacrossx‐axis
(f) reflectionandstretchbyfactorof
2acrossx‐axis.
(c) v.a.:
(d) v.a.:
(e) missingpointatx=3;x‐int:none;y‐int:(0,9)
(f) v.a.:x=−2;o.a.:y=2x−5;x‐int:
(g) v.a.:x= ;h.a.:
(a) v.a.:x=1,x=−1;h.a.:y=0;x‐int:none;y‐int:(0,−1)
(b) nopointsofdiscontinuity;h.a.:y=3;x‐intandy‐int:(0,0)
,x=−1;h.a.:
;x‐int:(−2,0),(3,0);y‐int:(0,6)
,x=1;h.a.:y=0;x‐intandy‐int:(0,0)
;x‐int:
√
,0 ,
√
, 0 ;y‐int:(0,−2)
, 0 ;y‐int:(0,−4)
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