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Lesson: Simplifying Rational Functions

In this lesson, we will learn how to simplify rational functions and how to find their domains.

Sample Question Videos

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Worksheet • 25 Questions • 3 Videos

Q1:

Simplify the function , and find its domain.

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

Q2:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 + 1 𝑥 + 3 𝑥 + 2 2 and find its domain.

  • A 𝑛 ( 𝑥 ) = 1 𝑥 + 2 , domain = ℝ − { − 1 , − 2 }
  • B 𝑛 ( 𝑥 ) = 𝑥 + 1 ( 𝑥 − 1 ) ( 𝑥 − 2 ) , domain = ℝ − { 1 , 2 }
  • C 𝑛 ( 𝑥 ) = 1 𝑥 + 2 , domain = ℝ − { − 2 }
  • D 𝑛 ( 𝑥 ) = 1 𝑥 − 2 , domain = ℝ − { 2 }
  • E 𝑛 ( 𝑥 ) = 1 𝑥 − 2 , domain = ℝ − { − 1 , 2 }

Q3:

Given the function 𝑓 ( 𝑥 ) = 7 4 𝑥 − 8 1 + 1 9 𝑥 − 2 𝑥 2 2 , evaluate 𝑓 ( 3 ) .

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

Q4:

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

  • A 𝑓 ( 𝑥 ) = 𝑥 + 6 𝑥 + 7 , domain = ℝ − − 1 7 , − 7
  • B 𝑓 ( 𝑥 ) = 𝑥 − 6 𝑥 + 7 , domain = ℝ − − 1 7 , − 7
  • C 𝑓 ( 𝑥 ) = 𝑥 + 6 𝑥 + 7 , domain = ℝ − { − 7 }
  • D 𝑓 ( 𝑥 ) = 𝑥 − 6 𝑥 − 7 , domain = ℝ − { − 7 }
  • E 𝑓 ( 𝑥 ) = 𝑥 − 6 𝑥 − 7 , domain = ℝ − − 1 7 , − 7

Q5:

Given that 𝑛 ( 𝑥 ) = 𝑥 − 5 𝑥 + 5 1 and 𝑛 ( 𝑥 ) = 𝑥 − 5 𝑥 𝑥 + 5 𝑥 2 2 2 , find the largest set on which the functions 𝑛 1 and 𝑛 2 are equal.

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

Q6:

Given the functions 𝑛 ( 𝑥 ) = 𝑥 𝑥 − 1 0 𝑥 1 2 and 𝑛 ( 𝑥 ) = 1 𝑥 − 1 0 2 , what is the set of values on which 𝑛 = 𝑛 1 2 ?

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

Q7:

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

  • Athe domain of 𝑛 = 1 the domain of 𝑛 2 and 𝑛 ( 𝑥 ) = 𝑛 ( 𝑥 ) 1 2 for each 𝑥 in the common domain
  • B the domain of 𝑛 = 1 the domain of 𝑛 2
  • C the domain of 𝑛 = 1 the domain of 𝑛 2 and 𝑛 ( 𝑥 ) ≠ 𝑛 ( 𝑥 ) 1 2
  • D 𝑛 ( 𝑥 ) = 𝑛 ( 𝑥 ) 1 2

Q8:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 + 1 ( 𝑥 + 1 ) ( 𝑥 − 3 ) 3 and find its domain.

  • A 𝑛 ( 𝑥 ) = 𝑥 − 𝑥 + 1 𝑥 − 3 2 , domain = ℝ − { − 1 , 3 }
  • B 𝑛 ( 𝑥 ) = 𝑥 + 𝑥 + 1 𝑥 − 3 2 , domain = ℝ − { − 1 , 3 }
  • C 𝑛 ( 𝑥 ) = 𝑥 − 𝑥 + 1 𝑥 − 3 2 , domain = ℝ − { 3 }
  • D 𝑛 ( 𝑥 ) = ( 𝑥 + 1 ) 𝑥 − 3 2 , domain = ℝ − { 3 }
  • E 𝑛 ( 𝑥 ) = ( 𝑥 + 1 ) 𝑥 − 3 2 , domain = ℝ − { − 1 , 3 }

Q9:

Simplify the function 𝑓 ( 𝑥 ) = 𝑥 − 8 1 𝑥 + 7 2 9 2 3 and find its domain.

  • A 𝑓 ( 𝑥 ) = 𝑥 − 9 𝑥 − 9 𝑥 + 8 1 2 , domain = ℝ − { − 9 }
  • B 𝑓 ( 𝑥 ) = 𝑥 − 9 𝑥 − 9 𝑥 + 8 1 2 , domain = ℝ − { 9 }
  • C 𝑓 ( 𝑥 ) = 𝑥 − 9 𝑥 − 9 𝑥 + 8 1 2 , domain = ℝ
  • D 𝑓 ( 𝑥 ) = 𝑥 + 9 𝑥 + 9 𝑥 + 8 1 2 , domain = ℝ
  • E 𝑓 ( 𝑥 ) = 𝑥 + 9 𝑥 + 9 𝑥 + 8 1 2 , domain = ℝ − { − 9 }

Q10:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 − 1 2 5 𝑥 + 5 𝑥 + 2 5 6 4 2 , and find its domain.

  • A 𝑛 ( 𝑥 ) = 𝑥 − √ 5 𝑥 + √ 5 , domain = ℝ
  • B 𝑛 ( 𝑥 ) = 𝑥 − 5 3 , domain = ℝ
  • C 𝑛 ( 𝑥 ) = 𝑥 − √ 5 𝑥 + √ 5 , domain = ℝ − { 5 }
  • D 𝑛 ( 𝑥 ) = 𝑥 − 5 2 , domain = ℝ − { − 5 }
  • E 𝑛 ( 𝑥 ) = 𝑥 − 5 2 , domain = ℝ

Q11:

Simplify the function 𝑓 ( 𝑥 ) = 𝑥 + 𝑥 − 8 0 𝑥 − 4 3 2 , and find its domain.

  • A 𝑥 + 5 𝑥 + 2 0 2 , domain = ℝ − { 4 }
  • B 𝑥 + 4 𝑥 + 2 0 2 , domain = ℝ − { 4 }
  • C 𝑥 + 5 𝑥 + 2 0 2 , domain = ℝ
  • D 𝑥 − 3 𝑥 + 2 0 2 , domain = ℝ
  • E 𝑥 − 3 𝑥 + 2 0 2 , domain = ℝ − { 4 }

Q12:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 + 𝑥 − 2 0 𝑥 + 5 𝑥 − 1 6 𝑥 − 8 0 2 3 2 , and find its domain.

  • A 𝑛 ( 𝑥 ) = 1 𝑥 + 4 , domain = ℝ − { − 5 , − 4 , 4 }
  • B 𝑛 ( 𝑥 ) = 𝑥 − 4 𝑥 + 1 6 2 , domain = ℝ − { − 5 , − 4 , 4 }
  • C 𝑛 ( 𝑥 ) = 1 𝑥 + 4 , domain = ℝ − { 4 }
  • D 𝑛 ( 𝑥 ) = 1 𝑥 − 4 , domain = ℝ − { − 4 }
  • E 𝑛 ( 𝑥 ) = 1 𝑥 − 4 , domain = ℝ − { − 5 , − 4 , 4 }

Q13:

Given that the algebraic fraction 𝑛 ( 𝑥 ) = 8 𝑥 ( 𝑥 + 4 ) 𝑥 + 𝑎 simplifies to 𝑛 ( 𝑥 ) = 8 𝑥 , what is the value of 𝑎 ?

Q14:

Given that 𝑛 ( 𝑥 ) = 𝑥 + 6 4 𝑥 − 1 6 1 , 𝑛 ( 𝑥 ) = 4 𝑥 + 2 5 6 𝑥 − 1 6 2 , and 𝑛 ( 𝑥 ) = 𝑛 ( 𝑥 ) ÷ 𝑛 ( 𝑥 ) 1 2 , find 𝑛 ( − 4 ) if possible.

  • A 1 4
  • B64
  • C4
  • D 1 2
  • E 1 6 4

Q15:

Given that 𝑛 ( 𝑥 ) = 𝑥 + 1 2 𝑥 + 3 6 𝑥 − 𝑎 2 2 simplifies to 𝑛 ( 𝑥 ) = 𝑥 + 6 𝑥 − 6 , what is the value of 𝑎 ?

Q16:

Simplify the function , and find its domain.

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

Q17:

Given that the functions 𝑛 ( 𝑥 ) = 8 𝑥 𝑥 + 𝑐 1 and 𝑛 ( 𝑥 ) = 8 𝑥 + 𝑑 𝑥 𝑥 + 𝑐 𝑥 + 5 𝑥 − 1 5 2 3 3 2 are equal, what are the values of 𝑐 and 𝑑 ?

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

Q18:

Which of the following functions are equal?

  • A 𝑛 ( 𝑥 ) = 𝑥 − 7 2 9 𝑥 + 9 𝑥 + 8 1 𝑥 , 𝑛 ( 𝑥 ) = ( 𝑥 − 9 ) 𝑥 + 6 3 𝑥 + 6 3 𝑥
  • B 𝑛 ( 𝑥 ) = 𝑥 − 7 2 9 𝑥 + 9 𝑥 + 8 1 𝑥 , 𝑛 ( 𝑥 ) = ( 𝑥 − 9 ) ( 𝑥 − 6 3 ) 𝑥 − 6 3 𝑥
  • C 𝑛 ( 𝑥 ) = 𝑥 − 7 2 9 𝑥 + 9 𝑥 + 8 1 𝑥 , 𝑛 ( 𝑥 ) = ( 𝑥 − 9 ) 𝑥 − 6 3 𝑥 − 6 3 𝑥
  • D 𝑛 ( 𝑥 ) = 𝑥 − 7 2 9 𝑥 + 9 𝑥 + 8 1 𝑥 , 𝑛 ( 𝑥 ) = ( 𝑥 − 9 ) ( 𝑥 + 6 3 ) 𝑥 + 6 3 𝑥
  • E 𝑛 ( 𝑥 ) = 𝑥 − 7 2 9 𝑥 + 9 𝑥 + 8 1 𝑥 , 𝑛 ( 𝑥 ) = ( 𝑥 − 9 ) ( 𝑥 + 6 3 ) 𝑥 + 6 3 𝑥

Q19:

Simplify the function and find its domain.

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

Q20:

Which of the following functions are equal?

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

Q21:

Given that 𝑛 ( 𝑥 ) = 𝑥 − 𝑎 𝑥 − 3 2 𝑥 + 𝑥 − 7 2 2 2 , and the multiplicative inverse of 𝑛 is 𝑥 + 9 𝑥 + 4 , what is the value of 𝑎 ?

Q22:

Given the functions 𝑝 ( 𝑥 ) = 3 𝑥 − 3 0 𝑥 ( 𝑥 + 1 0 ) ( 𝑥 − 1 0 ) 2 and 𝑞 ( 𝑥 ) = 3 𝑥 𝑥 + 1 0 , what is the set of values on which 𝑝 = 𝑞 ?

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

Q23:

Given that the multiplicative inverse of the function 𝑛 ( 𝑥 ) = 2 𝑥 + 1 0 𝑥 𝑥 + 1 4 𝑥 + 𝑎 2 2 is 𝑥 + 9 2 𝑥 , find the value of 𝑎 .

Q24:

Determine the domain of the function 𝑛 ( 𝑥 ) = 𝑥 − 6 4 8 𝑥 + 7 𝑥 ÷ 9 𝑥 − 1 1 7 𝑥 + 3 6 0 6 4 𝑥 − 4 9 2 2 2 2 .

  • A ℝ − − 7 8 , 0 , 7 8 , 5 , 8
  • B ℝ − { 0 , 5 }
  • C ℝ − − 7 8 , 7 8 , 5 , 8
  • D ℝ − − 7 8 , 0 , 5 , 8
  • E ℝ − − 7 8 , 0 , 7 8

Q25:

Simplify the function 𝑓 ( 𝑥 ) = 𝑥 − 4 𝑥 − 𝑥 − 2 2 2 and find its domain.

  • A 𝑓 ( 𝑥 ) = 𝑥 + 2 𝑥 + 1 , domain = ℝ − { 2 , − 1 }
  • B 𝑓 ( 𝑥 ) = − 4 − 𝑥 − 2 , domain = ℝ − { 2 , − 1 }
  • C 𝑓 ( 𝑥 ) = 𝑥 + 2 𝑥 + 1 , domain = ℝ − { − 1 }
  • D 𝑓 ( 𝑥 ) = 𝑥 − 2 𝑥 − 1 , domain = ℝ − { 1 }
  • E 𝑓 ( 𝑥 ) = 𝑥 − 2 𝑥 − 1 , domain = ℝ − { − 2 , 1 }
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