Worksheet: Simplifying Rational Functions

In this worksheet, we will practice simplifying rational functions and finding their domains.

Q1:

Simplify the function ๐‘“(๐‘ฅ)=๐‘ฅ+2๐‘ฅ๐‘ฅโˆ’4๏Šจ๏Šจ, and find its domain.

  • A๐‘“(๐‘ฅ)=๐‘ฅ๐‘ฅโˆ’2, domain =โ„โˆ’{2}
  • B๐‘“(๐‘ฅ)=๐‘ฅ๐‘ฅ+2, domain =โ„โˆ’{โˆ’2}
  • C๐‘“(๐‘ฅ)=๐‘ฅ๐‘ฅโˆ’2, domain =โ„โˆ’{โˆ’2,2}
  • D๐‘“(๐‘ฅ)=๐‘ฅ๐‘ฅ+2, domain =โ„โˆ’{โˆ’2,2}
  • E๐‘“(๐‘ฅ)=๐‘ฅ+2๐‘ฅ(๐‘ฅ+2)(๐‘ฅโˆ’2)๏Šจ, domain =โ„โˆ’{โˆ’2,2}

Q2:

Simplify the function ๐‘›(๐‘ฅ)=๐‘ฅ+1๐‘ฅ+3๐‘ฅ+2๏Šจ and find its domain.

  • A๐‘›(๐‘ฅ)=๐‘ฅ+1(๐‘ฅโˆ’1)(๐‘ฅโˆ’2), domain =โ„โˆ’{1,2}
  • B๐‘›(๐‘ฅ)=1๐‘ฅโˆ’2, domain =โ„โˆ’{โˆ’1,2}
  • C๐‘›(๐‘ฅ)=1๐‘ฅ+2, domain =โ„โˆ’{โˆ’2}
  • D๐‘›(๐‘ฅ)=1๐‘ฅโˆ’2, domain =โ„โˆ’{2}
  • E๐‘›(๐‘ฅ)=1๐‘ฅ+2, domain =โ„โˆ’{โˆ’1,โˆ’2}

Q3:

Given the function ๐‘“(๐‘ฅ)=74๐‘ฅโˆ’81+19๐‘ฅโˆ’2๐‘ฅ๏Šจ๏Šจ, evaluate ๐‘“(3).

  • A๐‘“(3)=โˆ’245
  • B๐‘“(3)=โˆ’29
  • C๐‘“(3)=3445
  • D๐‘“(3)=845

Q4:

Simplify the function ๐‘“(๐‘ฅ)=7๐‘ฅ+43๐‘ฅ+67๐‘ฅ+50๐‘ฅ+7๏Šจ๏Šจ, and find its domain.

  • A๐‘“(๐‘ฅ)=๐‘ฅโˆ’6๐‘ฅโˆ’7, domain =โ„โˆ’๏ฌโˆ’17,โˆ’7๏ธ
  • B๐‘“(๐‘ฅ)=๐‘ฅ+6๐‘ฅ+7, domain =โ„โˆ’{โˆ’7}
  • C๐‘“(๐‘ฅ)=๐‘ฅโˆ’6๐‘ฅโˆ’7, domain =โ„โˆ’{โˆ’7}
  • D๐‘“(๐‘ฅ)=๐‘ฅ+6๐‘ฅ+7, domain =โ„โˆ’๏ฌโˆ’17,โˆ’7๏ธ
  • E๐‘“(๐‘ฅ)=๐‘ฅโˆ’6๐‘ฅ+7, domain =โ„โˆ’๏ฌโˆ’17,โˆ’7๏ธ

Q5:

Given that ๐‘›(๐‘ฅ)=๐‘ฅโˆ’5๐‘ฅ+5๏Šง and ๐‘›(๐‘ฅ)=๐‘ฅโˆ’5๐‘ฅ๐‘ฅ+5๐‘ฅ๏Šจ๏Šจ๏Šจ, find the largest set on which the functions ๐‘›๏Šง and ๐‘›๏Šจ are equal.

  • Aโ„โˆ’{0,5}
  • Bโ„โˆ’{5}
  • Cโ„โˆ’{โˆ’5}
  • Dโ„โˆ’{โˆ’5,0,5}
  • Eโ„โˆ’{โˆ’5,0}

Q6:

Given the functions ๐‘›(๐‘ฅ)=๐‘ฅ๐‘ฅโˆ’10๐‘ฅ๏Šง๏Šจ and ๐‘›(๐‘ฅ)=1๐‘ฅโˆ’10๏Šจ, what is the set of values on which ๐‘›=๐‘›๏Šง๏Šจ?

  • Aโ„โˆ’{10}
  • Bโ„โˆ’{0,10}
  • Cโ„โˆ’{โˆ’10,0}
  • Dโ„โˆ’{0}
  • E{0}

Q7:

Which of the following statements describes when two functions ๐‘›๏Šง and ๐‘›๏Šจ are equal?

  • Athe domain of ๐‘›=๏Šง the domain of ๐‘›๏Šจ and ๐‘›(๐‘ฅ)=๐‘›(๐‘ฅ)๏Šง๏Šจ for each ๐‘ฅ in the common domain
  • B๐‘›(๐‘ฅ)=๐‘›(๐‘ฅ)๏Šง๏Šจ
  • Cthe domain of ๐‘›=๏Šง the domain of ๐‘›๏Šจ
  • Dthe domain of ๐‘›=๏Šง the domain of ๐‘›๏Šจ and ๐‘›(๐‘ฅ)โ‰ ๐‘›(๐‘ฅ)๏Šง๏Šจ

Q8:

Simplify the function ๐‘›(๐‘ฅ)=๐‘ฅ+1(๐‘ฅ+1)(๐‘ฅโˆ’3)๏Šฉ and find its domain.

  • A๐‘›(๐‘ฅ)=(๐‘ฅ+1)๐‘ฅโˆ’3๏Šจ, domain =โ„โˆ’{โˆ’1,3}
  • B๐‘›(๐‘ฅ)=๐‘ฅโˆ’๐‘ฅ+1๐‘ฅโˆ’3๏Šจ, domain =โ„โˆ’{3}
  • C๐‘›(๐‘ฅ)=๐‘ฅ+๐‘ฅ+1๐‘ฅโˆ’3๏Šจ, domain =โ„โˆ’{โˆ’1,3}
  • D๐‘›(๐‘ฅ)=๐‘ฅโˆ’๐‘ฅ+1๐‘ฅโˆ’3๏Šจ, domain =โ„โˆ’{โˆ’1,3}
  • E๐‘›(๐‘ฅ)=(๐‘ฅ+1)๐‘ฅโˆ’3๏Šจ, domain =โ„โˆ’{3}

Q9:

Simplify the function ๐‘“(๐‘ฅ)=๐‘ฅโˆ’81๐‘ฅ+729๏Šจ๏Šฉ and find its domain.

  • A๐‘“(๐‘ฅ)=๐‘ฅโˆ’9๐‘ฅโˆ’9๐‘ฅ+81๏Šจ, domain =โ„โˆ’{9}
  • B๐‘“(๐‘ฅ)=๐‘ฅ+9๐‘ฅ+9๐‘ฅ+81๏Šจ, domain =โ„โˆ’{โˆ’9}
  • C๐‘“(๐‘ฅ)=๐‘ฅโˆ’9๐‘ฅโˆ’9๐‘ฅ+81๏Šจ, domain =โ„
  • D๐‘“(๐‘ฅ)=๐‘ฅ+9๐‘ฅ+9๐‘ฅ+81๏Šจ, domain =โ„
  • E๐‘“(๐‘ฅ)=๐‘ฅโˆ’9๐‘ฅโˆ’9๐‘ฅ+81๏Šจ, domain =โ„โˆ’{โˆ’9}

Q10:

Simplify the function ๐‘›(๐‘ฅ)=๐‘ฅโˆ’125๐‘ฅ+5๐‘ฅ+25๏Šฌ๏Šช๏Šจ, and find its domain.

  • A๐‘›(๐‘ฅ)=๏€ป๐‘ฅโˆ’โˆš5๏‡๏€ป๐‘ฅ+โˆš5๏‡, domain =โ„โˆ’{5}
  • B๐‘›(๐‘ฅ)=๐‘ฅโˆ’5๏Šจ, domain =โ„โˆ’{โˆ’5}
  • C๐‘›(๐‘ฅ)=๐‘ฅโˆ’5๏Šจ, domain =โ„
  • D๐‘›(๐‘ฅ)=๐‘ฅโˆ’5๏Šฉ, domain =โ„
  • E๐‘›(๐‘ฅ)=๏€ป๐‘ฅโˆ’โˆš5๏‡๏€ป๐‘ฅ+โˆš5๏‡, domain =โ„

Q11:

Simplify the function ๐‘“(๐‘ฅ)=๐‘ฅ+๐‘ฅโˆ’80๐‘ฅโˆ’4๏Šฉ๏Šจ, and find its domain.

  • A๐‘ฅโˆ’3๐‘ฅ+20๏Šจ, domain =โ„โˆ’{4}
  • B๐‘ฅ+5๐‘ฅ+20๏Šจ, domain =โ„โˆ’{4}
  • C๐‘ฅ+5๐‘ฅ+20๏Šจ, domain =โ„
  • D๐‘ฅโˆ’3๐‘ฅ+20๏Šจ, domain =โ„
  • E๐‘ฅ+4๐‘ฅ+20๏Šจ, domain =โ„โˆ’{4}

Q12:

Simplify the function ๐‘›(๐‘ฅ)=๐‘ฅ+๐‘ฅโˆ’20๐‘ฅ+5๐‘ฅโˆ’16๐‘ฅโˆ’80๏Šจ๏Šฉ๏Šจ, and find its domain.

  • A๐‘›(๐‘ฅ)=1๐‘ฅ+4, domain =โ„โˆ’{4}
  • B๐‘›(๐‘ฅ)=1๐‘ฅโˆ’4, domain =โ„โˆ’{โˆ’4}
  • C๐‘›(๐‘ฅ)=1๐‘ฅโˆ’4, domain =โ„โˆ’{โˆ’5,โˆ’4,4}
  • D๐‘›(๐‘ฅ)=1๐‘ฅ+4, domain =โ„โˆ’{โˆ’5,โˆ’4,4}
  • E๐‘›(๐‘ฅ)=๐‘ฅโˆ’4๐‘ฅ+16๏Šจ, domain =โ„โˆ’{โˆ’5,โˆ’4,4}

Q13:

Given that the algebraic fraction ๐‘›(๐‘ฅ)=8๐‘ฅ(๐‘ฅ+4)๐‘ฅ+๐‘Ž simplifies to ๐‘›(๐‘ฅ)=8๐‘ฅ, what is the value of ๐‘Ž?

Q14:

Given that ๐‘›(๐‘ฅ)=๐‘ฅ+64๐‘ฅโˆ’16๏Šง, ๐‘›(๐‘ฅ)=4๐‘ฅ+256๐‘ฅโˆ’16๏Šจ, and ๐‘›(๐‘ฅ)=๐‘›(๐‘ฅ)รท๐‘›(๐‘ฅ)๏Šง๏Šจ, find ๐‘›(โˆ’4) if possible.

  • A164
  • B12
  • C14
  • D64
  • E4

Q15:

Given that ๐‘›(๐‘ฅ)=๐‘ฅ+12๐‘ฅ+36๐‘ฅโˆ’๐‘Ž๏Šจ๏Šจ simplifies to ๐‘›(๐‘ฅ)=๐‘ฅ+6๐‘ฅโˆ’6, what is the value of ๐‘Ž?

Q16:

Simplify the function ๐‘“(๐‘ฅ)=(๐‘ฅ+3)โˆ’36๐‘ฅ(๐‘ฅโˆ’3)๏Šจ, and find its domain.

  • A๐‘“(๐‘ฅ)=๐‘ฅ+9๐‘ฅ, domain =โ„โˆ’{0,โˆ’3}
  • B๐‘“(๐‘ฅ)=๐‘ฅโˆ’9๐‘ฅ, domain =โ„โˆ’{0}
  • C๐‘“(๐‘ฅ)=๐‘ฅ+9๐‘ฅ, domain =โ„โˆ’{0,3}
  • D๐‘“(๐‘ฅ)=๐‘ฅ+9๐‘ฅ, domain =โ„โˆ’{0}
  • E๐‘“(๐‘ฅ)=๐‘ฅโˆ’9๐‘ฅ, domain =โ„โˆ’{0,3}

Q17:

Given that the functions ๐‘›(๐‘ฅ)=8๐‘ฅ๐‘ฅ+๐‘๏Šง and ๐‘›(๐‘ฅ)=8๐‘ฅ+๐‘‘๐‘ฅ๐‘ฅ+๐‘๐‘ฅ+5๐‘ฅโˆ’15๏Šจ๏Šฉ๏Šฉ๏Šจ are equal, what are the values of ๐‘ and ๐‘‘?

  • A๐‘=3, ๐‘‘=โˆ’40
  • B๐‘=โˆ’3, ๐‘‘=5
  • C๐‘=3, ๐‘‘=40
  • D๐‘=โˆ’3, ๐‘‘=โˆ’40
  • E๐‘=โˆ’3, ๐‘‘=40

Q18:

Which of the following functions are equal?

  • A๐‘›(๐‘ฅ)=๐‘ฅโˆ’729๐‘ฅ+9๐‘ฅ+81๐‘ฅ๏Šง๏Šฉ๏Šฉ๏Šจ, ๐‘›(๐‘ฅ)=(๐‘ฅโˆ’9)(๐‘ฅ+63)๐‘ฅ+63๐‘ฅ๏Šจ๏Šจ
  • B๐‘›(๐‘ฅ)=๐‘ฅโˆ’729๐‘ฅ+9๐‘ฅ+81๐‘ฅ๏Šง๏Šฉ๏Šฉ๏Šจ, ๐‘›(๐‘ฅ)=(๐‘ฅโˆ’9)๏€น๐‘ฅโˆ’63๏…๐‘ฅโˆ’63๐‘ฅ๏Šจ๏Šจ๏Šฉ
  • C๐‘›(๐‘ฅ)=๐‘ฅโˆ’729๐‘ฅ+9๐‘ฅ+81๐‘ฅ๏Šง๏Šฉ๏Šฉ๏Šจ, ๐‘›(๐‘ฅ)=(๐‘ฅโˆ’9)(๐‘ฅ+63)๐‘ฅ+63๐‘ฅ๏Šจ๏Šฉ
  • D๐‘›(๐‘ฅ)=๐‘ฅโˆ’729๐‘ฅ+9๐‘ฅ+81๐‘ฅ๏Šง๏Šฉ๏Šฉ๏Šจ, ๐‘›(๐‘ฅ)=(๐‘ฅโˆ’9)๏€น๐‘ฅ+63๏…๐‘ฅ+63๐‘ฅ๏Šจ๏Šจ๏Šฉ
  • E๐‘›(๐‘ฅ)=๐‘ฅโˆ’729๐‘ฅ+9๐‘ฅ+81๐‘ฅ๏Šง๏Šฉ๏Šฉ๏Šจ, ๐‘›(๐‘ฅ)=(๐‘ฅโˆ’9)(๐‘ฅโˆ’63)๐‘ฅโˆ’63๐‘ฅ๏Šจ๏Šฉ

Q19:

Simplify the function ๐‘“(๐‘ฅ)=๐‘ฅ+4๐‘ฅ+45๐‘ฅโˆ’20๐‘ฅ๏Šจ๏Šฉ and find its domain.

  • A๐‘“(๐‘ฅ)=๐‘ฅ+25๐‘ฅ(๐‘ฅโˆ’2), domain =โ„โˆ’{0,โˆ’2,2}
  • B๐‘“(๐‘ฅ)=(๐‘ฅ+2)5๐‘ฅ(๐‘ฅโˆ’4)๏Šจ๏Šจ, domain =โ„โˆ’{0,โˆ’2,2}
  • C๐‘“(๐‘ฅ)=๐‘ฅโˆ’25๐‘ฅ(๐‘ฅ+2), domain =โ„โˆ’{0,โˆ’2}
  • D๐‘“(๐‘ฅ)=๐‘ฅโˆ’25๐‘ฅ(๐‘ฅ+2), domain =โ„โˆ’{0,โˆ’2,2}
  • E๐‘“(๐‘ฅ)=๐‘ฅ+25๐‘ฅ(๐‘ฅโˆ’2), domain =โ„โˆ’{0,2}

Q20:

Which of the following functions are equal?

  • A๐‘›(๐‘ฅ)=๐‘ฅ+125๐‘ฅโˆ’5๐‘ฅ+25๐‘ฅ๏Šง๏Šฉ๏Šฉ๏Šจ, ๐‘›(๐‘ฅ)=๐‘ฅ+3๐‘ฅโˆ’10๐‘ฅโˆ’2๐‘ฅ๏Šจ๏Šจ๏Šฉ
  • B๐‘›(๐‘ฅ)=๐‘ฅ+125๐‘ฅโˆ’5๐‘ฅ+25๐‘ฅ๏Šง๏Šฉ๏Šฉ๏Šจ, ๐‘›(๐‘ฅ)=๐‘ฅ+7๐‘ฅ+10๐‘ฅ+2๐‘ฅ๏Šจ๏Šจ๏Šฉ
  • C๐‘›(๐‘ฅ)=๐‘ฅ+125๐‘ฅโˆ’5๐‘ฅ+25๐‘ฅ๏Šง๏Šฉ๏Šฉ๏Šจ, ๐‘›(๐‘ฅ)=๐‘ฅ+5๐‘ฅโˆ’2๐‘ฅโˆ’10๐‘ฅโˆ’2๐‘ฅ๏Šจ๏Šฉ๏Šจ๏Šฉ
  • D๐‘›(๐‘ฅ)=๐‘ฅ+125๐‘ฅโˆ’5๐‘ฅ+25๐‘ฅ๏Šง๏Šฉ๏Šฉ๏Šจ, ๐‘›(๐‘ฅ)=๐‘ฅ+7๐‘ฅ+10๐‘ฅ+2๐‘ฅ๏Šจ๏Šจ๏Šจ
  • E๐‘›(๐‘ฅ)=๐‘ฅ+125๐‘ฅโˆ’5๐‘ฅ+25๐‘ฅ๏Šง๏Šฉ๏Šฉ๏Šจ, ๐‘›(๐‘ฅ)=๐‘ฅ+5๐‘ฅ+2๐‘ฅ+10๐‘ฅ+2๐‘ฅ๏Šจ๏Šฉ๏Šจ๏Šฉ

Q21:

Given that ๐‘›(๐‘ฅ)=๐‘ฅโˆ’๐‘Ž๐‘ฅโˆ’32๐‘ฅ+๐‘ฅโˆ’72๏Šจ๏Šจ, and the multiplicative inverse of ๐‘› is ๐‘ฅ+9๐‘ฅ+4, what is the value of ๐‘Ž?

Q22:

Given the functions ๐‘(๐‘ฅ)=3๐‘ฅโˆ’30๐‘ฅ(๐‘ฅ+10)(๐‘ฅโˆ’10)๏Šจ and ๐‘ž(๐‘ฅ)=3๐‘ฅ๐‘ฅ+10, what is the set of values on which ๐‘=๐‘ž?

  • Aโ„โˆ’{10,โˆ’10}
  • Bโ„โˆ’{โˆ’10,0}
  • Cโ„โˆ’{10}
  • Dโ„โˆ’{0,10}
  • Eโ„โˆ’{โˆ’10}

Q23:

Given that the multiplicative inverse of the function ๐‘›(๐‘ฅ)=2๐‘ฅ+10๐‘ฅ๐‘ฅ+14๐‘ฅ+๐‘Ž๏Šจ๏Šจ is ๐‘ฅ+92๐‘ฅ, find the value of ๐‘Ž.

Q24:

Determine the domain of the function ๐‘›(๐‘ฅ)=๐‘ฅโˆ’648๐‘ฅ+7๐‘ฅรท9๐‘ฅโˆ’117๐‘ฅ+36064๐‘ฅโˆ’49๏Šจ๏Šจ๏Šจ๏Šจ.

  • Aโ„โˆ’๏ฌโˆ’78,0,78๏ธ
  • Bโ„โˆ’{0,5}
  • Cโ„โˆ’๏ฌโˆ’78,0,5,8๏ธ
  • Dโ„โˆ’๏ฌโˆ’78,78,5,8๏ธ
  • Eโ„โˆ’๏ฌโˆ’78,0,78,5,8๏ธ

Q25:

Simplify the function ๐‘“(๐‘ฅ)=๐‘ฅโˆ’4๐‘ฅโˆ’๐‘ฅโˆ’2๏Šจ๏Šจ and find its domain.

  • A๐‘“(๐‘ฅ)=๐‘ฅ+2๐‘ฅ+1, domain =โ„โˆ’{โˆ’1}
  • B๐‘“(๐‘ฅ)=๐‘ฅโˆ’2๐‘ฅโˆ’1, domain =โ„โˆ’{1}
  • C๐‘“(๐‘ฅ)=โˆ’4โˆ’๐‘ฅโˆ’2, domain =โ„โˆ’{2,โˆ’1}
  • D๐‘“(๐‘ฅ)=๐‘ฅ+2๐‘ฅ+1, domain =โ„โˆ’{2,โˆ’1}
  • E๐‘“(๐‘ฅ)=๐‘ฅโˆ’2๐‘ฅโˆ’1, domain =โ„โˆ’{โˆ’2,1}

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