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MATH1151:ALGEBRA
FINALEXAMINATIONOUTLINE
ThefinalExamisscheduledfor:
Date:TuesdayMay14
th
2024
Place:SM-D1/SM-D2
Time:2.5hours8:00AM–10:30AM
THISEXAMPAPERCONSISTSOFEIGHT(8)QUESTIONSOFWHICHALLITEMS
MUSTBEANSWERED.TOTALPOINTS:100.SHOWALLNECESSARYWORKINGTO
OBTAINFULLMARKS.
Thefinalexaminationwilltakeonthefollowingstructure:
1.Simplifythefollowing:[24points]
a.RadicalExpression
Example:
38+218−42
b.Rationalexpressionsusinglawsofexponents
Example:
3
1
()
3
× 3
2
()
2
3
2
× 3
1
()
2
c.ComplexRationalExpression
Example:
푥
2
−1
3푥+2
푥+1
3푥
2
−푥−2
 

2.Usesyntheticdivisionorthefactorandremaindertheoremtoanswerthe
followingquestion:[6points]
Example:
Isafactorof?Statewhyitisorisnotafactor
푥−1푥
3
−5푥
2
+9푥−5
3.Solveforandinthesystemsofequations,[10points]
푥,푦푧
Example:
   푥+푦−푧=−1
4푥−3푦+2푧=16
2푥−2푦−3푧=5
4.Solvetheinequalitiesexpressingthesolutionsetinintervalnotationform:[20points]
Examples:
a.
5−2푥||−1≤2
b.
푥
2
+3푥−10<0
5.Forthefollowingradicalequations,findtherealsolutionsandidentify
anyextraneoussolutions[10points]
Examples:
a.
3
2푥−4−2=0
b.
5
푥
2
+2푥= −1

6.Decomposethefollowingtodeterminethepartialfractions:[10points]
Example:
4푥
(푥−2)(푥+1)
7.Findingtheinverseofafunction:[10points]
Example:
Verifythatinverseofis
푓(푥)=3푥+4 푓
−1
(푥)=
푥−4
3
Thefollowingarethestepstofindtheinverseofafunction:
Given
푓(푥)=3푥+4
1.Replacef(x)withy
푓(푥)=3푥+4
푦=3푥+4
2.Swapthexvariableswiththeyvariables
푥=3푦+4
3.Rearrangethefunctiontomakethevariableythesubjectoftheformula:
Subtract4frombothsides
푥=3푦+4
−4 −4
Dividebothsidesby
푥−4
3
=
3푦
3
3
푥−4
3
=푦
So,
푓
−1
(푥)=
푥−4
3
Therefore,theinverseofis
푓(푥)=3푥+4 푓
−1
(푥)=
푥−4
3

Example2:
Verifythatinverseofis
푓(푥)=7푥−4 푓
−1
(푥)=
푥+4
7
Thefollowingarethestepstofindtheinverseofafunction:
Given
푓(푥)=7푥−4
4.Replacef(x)withy
푓(푥)=7푥−4
푦=7푥−4
5.Swapthexvariableswiththeyvariables
푥=7푦−4
6.Rearrangethefunctiontomakethevariableythesubjectoftheformula:
Add4frombothsides
푥=7푦−4
+4 +4
Dividebothsidesby
푥+4
7
=
7푦
7
7
푥+4
7
=푦
So,
푓
−1
(푥)=
푥+4
7
Therefore,theinverseofis
푓(푥)=3푥+4 푓
−1
(푥)=
푥−4
3
HelpfulResources:
https://www.youtube.com/results?search_query=finding+the+inverse+of+a+funcitons
https://www.youtube.com/watch?v=2zeYEx4eTdc
8.Functionproperties
Givenfind[10points]
푦=푥
2
−4푥+3, 
a.andintercepts
푥푦 

b.vertex
c.statedomainandrange
d.Sketchthegraphandlabelthepointsfrompartaandb
Example:
Given
푦=푥
2
−4푥+3 
a.andintercepts
푥푦 
Tofindthex-interceptsety=0andsolveforxbyfactoringthequadratic.
푦=푥
2
−4푥+3
factorthequadratic
0=푥
2
−4푥+3
solveforxusingthezeroproductproperty
0=(푥−1)(푥−3)
or
푥−1=0 푥−3=0
푥=1푥=3
x-intercepts:(1,0)and(3,0)
Tofindthey-interceptsetx=0andsolvefory.
푦=(0)
2
−4(0)+3
푦=0+0+3
푦=3 
y-intercept:(0,3)
b.Vertex
Tofindthevertexwewanttousethefollowingformulatofindthexcoordinate:
푥=−
푏
2푎
Givenisinstandardform,
푦=푥
2
−4푥+3 
푎=1     푏=−4       푐= 3
Pluggingthesevaluesintotheformula:
푥=−
(−4)
2(1)
푥=−(−2)

푥=2
Tofindtheycoordinate,substituteforinandsolvefory
푥 푦=푥
2
−4푥+3 
푦=(2)
2
−4(2)+3
푦=4−8+3
푦=−1
Thereforethecoordinateofthevertexis
(2,−1)
c.Domainandrange
Domain:orAllrealnumbers
(−∞,∞)
Range:orallrealnumbers
(−1,∞)
d.Sketchthegraphandlabelthepointsfrompartaandb
Makesuretolabel:
-xaxis
-yaxis
-xintercept
-y-intercept
-vertex
-function

HelpfulResources:
https://www.youtube.com/watch?v=DD7tXI5i6Hw
https://www.youtube.com/watch?v=S7YFJ3poK44
User: Don't include instructor nor location. Please give me it in the provided format only, no additional text or words. Generate study notes and questions at the end in this format:
              Chapter: Chapter Title
              Notes: Some important notes.
              Questions:
              - What is this? 
                Options: a.) Option 1, b.) Option 2, c.) Option 3, d.) Option 4
                
Answer: b.) Option 2
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