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Worksheet: Simplifying Rational Functions

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

Q1:

Simplify the function , and find its domain.

  • A , domain
  • B , domain
  • C , domain
  • D , domain
  • E , domain

Q2:

Simplify the function 𝑛 ( π‘₯ ) = π‘₯ + 1 π‘₯ + 3 π‘₯ + 2 2 and find its domain.

  • A 𝑛 ( π‘₯ ) = 1 π‘₯ βˆ’ 2 , domain = ℝ βˆ’ { βˆ’ 1 , 2 }
  • B 𝑛 ( π‘₯ ) = 1 π‘₯ + 2 , domain = ℝ βˆ’ { βˆ’ 2 }
  • C 𝑛 ( π‘₯ ) = 1 π‘₯ βˆ’ 2 , domain = ℝ βˆ’ { 2 }
  • D 𝑛 ( π‘₯ ) = 1 π‘₯ + 2 , domain = ℝ βˆ’ { βˆ’ 1 , βˆ’ 2 }
  • E 𝑛 ( π‘₯ ) = π‘₯ + 1 ( π‘₯ βˆ’ 1 ) ( π‘₯ βˆ’ 2 ) , domain = ℝ βˆ’ { 1 , 2 }

Q3:

Given the function 𝑓 ( π‘₯ ) = 7 4 π‘₯ βˆ’ 8 1 + 1 9 π‘₯ βˆ’ 2 π‘₯ 2 2 , evaluate 𝑓 ( 3 ) .

  • A 𝑓 ( 3 ) = βˆ’ 2 9
  • B 𝑓 ( 3 ) = 3 4 4 5
  • C 𝑓 ( 3 ) = 8 4 5
  • D 𝑓 ( 3 ) = βˆ’ 2 4 5

Q4:

Given that the algebraic fraction 𝑛 ( π‘₯ ) = 8 π‘₯ ( π‘₯ + 4 ) π‘₯ + π‘Ž simplifies to 𝑛 ( π‘₯ ) = 8 π‘₯ , what is the value of π‘Ž ?

Q5:

Given that 𝑛 ( π‘₯ ) = π‘₯ + 1 2 π‘₯ + 3 6 π‘₯ βˆ’ π‘Ž 2 2 simplifies to 𝑛 ( π‘₯ ) = π‘₯ + 6 π‘₯ βˆ’ 6 , what is the value of π‘Ž ?

Q6:

Simplify the function 𝑛 ( π‘₯ ) = π‘₯ + 1 ( π‘₯ + 1 ) ( π‘₯ βˆ’ 3 ) 3 and find its domain.

  • A 𝑛 ( π‘₯ ) = ( π‘₯ + 1 ) π‘₯ βˆ’ 3 2 , domain = ℝ βˆ’ { βˆ’ 1 , 3 }
  • B 𝑛 ( π‘₯ ) = π‘₯ βˆ’ π‘₯ + 1 π‘₯ βˆ’ 3 2 , domain = ℝ βˆ’ { 3 }
  • C 𝑛 ( π‘₯ ) = ( π‘₯ + 1 ) π‘₯ βˆ’ 3 2 , domain = ℝ βˆ’ { 3 }
  • D 𝑛 ( π‘₯ ) = π‘₯ βˆ’ π‘₯ + 1 π‘₯ βˆ’ 3 2 , domain = ℝ βˆ’ { βˆ’ 1 , 3 }
  • E 𝑛 ( π‘₯ ) = π‘₯ + π‘₯ + 1 π‘₯ βˆ’ 3 2 , domain = ℝ βˆ’ { βˆ’ 1 , 3 }

Q7:

Simplify the function , and find its domain.

  • A , domain
  • B , domain
  • C , domain
  • D , domain
  • E , domain

Q8:

Given that the functions 𝑛 ( π‘₯ ) = 8 π‘₯ π‘₯ + 𝑐 1 and 𝑛 ( π‘₯ ) = 8 π‘₯ + 𝑑 π‘₯ π‘₯ + 𝑐 π‘₯ + 5 π‘₯ βˆ’ 1 5 2 3 3 2 are equal, what are the values of 𝑐 and 𝑑 ?

  • A 𝑐 = βˆ’ 3 , 𝑑 = βˆ’ 4 0
  • B 𝑐 = 3 , 𝑑 = 4 0
  • C 𝑐 = βˆ’ 3 , 𝑑 = 5
  • D 𝑐 = βˆ’ 3 , 𝑑 = 4 0
  • E 𝑐 = 3 , 𝑑 = βˆ’ 4 0

Q9:

Which of the following functions are equal?

  • A 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 7 2 9 π‘₯ + 9 π‘₯ + 8 1 π‘₯     , 𝑛 ( π‘₯ ) = ( π‘₯ βˆ’ 9 ) ( π‘₯ + 6 3 ) π‘₯ + 6 3 π‘₯  
  • B 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 7 2 9 π‘₯ + 9 π‘₯ + 8 1 π‘₯     , 𝑛 ( π‘₯ ) = ( π‘₯ βˆ’ 9 ) ο€Ή π‘₯ βˆ’ 6 3  π‘₯ βˆ’ 6 3 π‘₯   
  • C 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 7 2 9 π‘₯ + 9 π‘₯ + 8 1 π‘₯     , 𝑛 ( π‘₯ ) = ( π‘₯ βˆ’ 9 ) ( π‘₯ + 6 3 ) π‘₯ + 6 3 π‘₯  
  • D 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 7 2 9 π‘₯ + 9 π‘₯ + 8 1 π‘₯     , 𝑛 ( π‘₯ ) = ( π‘₯ βˆ’ 9 ) ο€Ή π‘₯ + 6 3  π‘₯ + 6 3 π‘₯   
  • E 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 7 2 9 π‘₯ + 9 π‘₯ + 8 1 π‘₯     , 𝑛 ( π‘₯ ) = ( π‘₯ βˆ’ 9 ) ( π‘₯ βˆ’ 6 3 ) π‘₯ βˆ’ 6 3 π‘₯  

Q10:

Simplify the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 8 1 π‘₯ + 7 2 9 2 3 and find its domain.

  • A 𝑓 ( π‘₯ ) = π‘₯ + 9 π‘₯ + 9 π‘₯ + 8 1 2 , domain = ℝ βˆ’ { βˆ’ 9 }
  • B 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 9 π‘₯ βˆ’ 9 π‘₯ + 8 1 2 , domain = ℝ
  • C 𝑓 ( π‘₯ ) = π‘₯ + 9 π‘₯ + 9 π‘₯ + 8 1 2 , domain = ℝ
  • D 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 9 π‘₯ βˆ’ 9 π‘₯ + 8 1 2 , domain = ℝ βˆ’ { βˆ’ 9 }
  • E 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 9 π‘₯ βˆ’ 9 π‘₯ + 8 1 2 , domain = ℝ βˆ’ { 9 }

Q11:

Simplify the function and find its domain.

  • A , domain
  • B , domain
  • C , domain
  • D , domain
  • E , domain

Q12:

Which of the following functions are equal?

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q13:

Given that 𝑛 ( π‘₯ ) = π‘₯ βˆ’ π‘Ž π‘₯ βˆ’ 3 2 π‘₯ + π‘₯ βˆ’ 7 2 2 2 , and the multiplicative inverse of 𝑛 is π‘₯ + 9 π‘₯ + 4 , what is the value of π‘Ž ?

Q14:

Simplify the function 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 1 2 5 π‘₯ + 5 π‘₯ + 2 5 6 4 2 , and find its domain.

  • A 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 5 2 , domain = ℝ
  • B 𝑛 ( π‘₯ ) = ο€» π‘₯ βˆ’ √ 5  ο€» π‘₯ + √ 5  , domain = ℝ βˆ’ { 5 }
  • C 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 5 2 , domain = ℝ βˆ’ { βˆ’ 5 }
  • D 𝑛 ( π‘₯ ) = ο€» π‘₯ βˆ’ √ 5  ο€» π‘₯ + √ 5  , domain = ℝ
  • E 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 5 3 , domain = ℝ

Q15:

Given the functions 𝑝 ( π‘₯ ) = 3 π‘₯ βˆ’ 3 0 π‘₯ ( π‘₯ + 1 0 ) ( π‘₯ βˆ’ 1 0 ) 2 and π‘ž ( π‘₯ ) = 3 π‘₯ π‘₯ + 1 0 , what is the set of values on which 𝑝 = π‘ž ?

  • A ℝ βˆ’ { 1 0 }
  • B ℝ βˆ’ { βˆ’ 1 0 }
  • C ℝ βˆ’ { 0 , 1 0 }
  • D ℝ βˆ’ { 1 0 , βˆ’ 1 0 }
  • E ℝ βˆ’ { βˆ’ 1 0 , 0 }

Q16:

Given that the multiplicative inverse of the function 𝑛 ( π‘₯ ) = 2 π‘₯ + 1 0 π‘₯ π‘₯ + 1 4 π‘₯ + π‘Ž 2 2 is π‘₯ + 9 2 π‘₯ , find the value of π‘Ž .

Q17:

Determine the domain of the function 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 6 4 8 π‘₯ + 7 π‘₯ Γ· 9 π‘₯ βˆ’ 1 1 7 π‘₯ + 3 6 0 6 4 π‘₯ βˆ’ 4 9 2 2 2 2 .

  • A ℝ βˆ’  βˆ’ 7 8 , 0 , 7 8 
  • B ℝ βˆ’  βˆ’ 7 8 , 7 8 , 5 , 8 
  • C ℝ βˆ’  βˆ’ 7 8 , 0 , 5 , 8 
  • D ℝ βˆ’  βˆ’ 7 8 , 0 , 7 8 , 5 , 8 
  • E ℝ βˆ’ { 0 , 5 }

Q18:

Given that 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 5 π‘₯ + 5 1 and 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 5 π‘₯ π‘₯ + 5 π‘₯ 2 2 2 , find the largest set on which the functions 𝑛 1 and 𝑛 2 are equal.

  • A ℝ βˆ’ { βˆ’ 5 }
  • B ℝ βˆ’ { 0 , 5 }
  • C ℝ βˆ’ { 5 }
  • D ℝ βˆ’ { βˆ’ 5 , 0 }
  • E ℝ βˆ’ { βˆ’ 5 , 0 , 5 }

Q19:

Given that 𝑛 ( π‘₯ ) = π‘₯ + 6 4 π‘₯ βˆ’ 1 6 1 , 𝑛 ( π‘₯ ) = 4 π‘₯ + 2 5 6 π‘₯ βˆ’ 1 6 2 , and 𝑛 ( π‘₯ ) = 𝑛 ( π‘₯ ) Γ· 𝑛 ( π‘₯ ) 1 2 , find 𝑛 ( βˆ’ 4 ) if possible.

  • A 1 6 4
  • B4
  • C 1 2
  • D 1 4
  • E64

Q20:

Given the functions 𝑛 ( π‘₯ ) = π‘₯ π‘₯ βˆ’ 1 0 π‘₯ 1 2 and 𝑛 ( π‘₯ ) = 1 π‘₯ βˆ’ 1 0 2 , what is the set of values on which 𝑛 = 𝑛 1 2 ?

  • A ℝ βˆ’ { 1 0 }
  • B ℝ βˆ’ { βˆ’ 1 0 , 0 }
  • C ℝ βˆ’ { 0 }
  • D ℝ βˆ’ { 0 , 1 0 }
  • E { 0 }

Q21:

Given the functions 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 1 π‘₯ βˆ’ 1 1 2 and 𝑛 ( π‘₯ ) = π‘₯ + 1 2 , what is the set of values on which 𝑛 = 𝑛 1 2 ?

  • A ℝ βˆ’ { βˆ’ 1 }
  • B ℝ βˆ’ { βˆ’ 1 , 1 }
  • C ℝ
  • D ℝ βˆ’ { 1 }
  • E { 1 }

Q22:

Simplify the function 𝑓 ( π‘₯ ) = π‘₯ + π‘₯ βˆ’ 8 0 π‘₯ βˆ’ 4 3 2 , and find its domain.

  • A π‘₯ βˆ’ 3 π‘₯ + 2 0 2 , domain = ℝ βˆ’ { 4 }
  • B π‘₯ + 5 π‘₯ + 2 0 2 , domain = ℝ
  • C π‘₯ βˆ’ 3 π‘₯ + 2 0 2 , domain = ℝ
  • D π‘₯ + 5 π‘₯ + 2 0 2 , domain = ℝ βˆ’ { 4 }
  • E π‘₯ + 4 π‘₯ + 2 0 2 , domain = ℝ βˆ’ { 4 }

Q23:

Simplify the function 𝑛 ( π‘₯ ) = π‘₯ + π‘₯ βˆ’ 2 0 π‘₯ + 5 π‘₯ βˆ’ 1 6 π‘₯ βˆ’ 8 0 2 3 2 , and find its domain.

  • A 𝑛 ( π‘₯ ) = 1 π‘₯ βˆ’ 4 , domain = ℝ βˆ’ { βˆ’ 5 , βˆ’ 4 , 4 }
  • B 𝑛 ( π‘₯ ) = 1 π‘₯ + 4 , domain = ℝ βˆ’ { 4 }
  • C 𝑛 ( π‘₯ ) = 1 π‘₯ βˆ’ 4 , domain = ℝ βˆ’ { βˆ’ 4 }
  • D 𝑛 ( π‘₯ ) = 1 π‘₯ + 4 , domain = ℝ βˆ’ { βˆ’ 5 , βˆ’ 4 , 4 }
  • E 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 4 π‘₯ + 1 6 2 , domain = ℝ βˆ’ { βˆ’ 5 , βˆ’ 4 , 4 }

Q24:

Simplify the function 𝑓 ( π‘₯ ) = 7 π‘₯ + 4 3 π‘₯ + 6 7 π‘₯ + 5 0 π‘₯ + 7 2 2 , and find its domain.

  • A 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 6 π‘₯ βˆ’ 7 , domain = ℝ βˆ’  βˆ’ 1 7 , βˆ’ 7 
  • B 𝑓 ( π‘₯ ) = π‘₯ + 6 π‘₯ + 7 , domain = ℝ βˆ’ { βˆ’ 7 }
  • C 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 6 π‘₯ βˆ’ 7 , domain = ℝ βˆ’ { βˆ’ 7 }
  • D 𝑓 ( π‘₯ ) = π‘₯ + 6 π‘₯ + 7 , domain = ℝ βˆ’  βˆ’ 1 7 , βˆ’ 7 
  • E 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 6 π‘₯ + 7 , domain = ℝ βˆ’  βˆ’ 1 7 , βˆ’ 7 

Q25:

Which of the following statements describes when two functions 𝑛 1 and 𝑛 2 are equal?

  • A 𝑛 ( π‘₯ ) = 𝑛 ( π‘₯ ) 1 2
  • B the domain of 𝑛 = 1 the domain of 𝑛 2
  • C the domain of 𝑛 = 1 the domain of 𝑛 2 and 𝑛 ( π‘₯ ) β‰  𝑛 ( π‘₯ ) 1 2
  • Dthe domain of 𝑛 = 1 the domain of 𝑛 2 and 𝑛 ( π‘₯ ) = 𝑛 ( π‘₯ ) 1 2 for each π‘₯ in the common domain

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