Worksheet: Adding and Subtracting Rational Functions

In this worksheet, we will practice adding and subtracting rational functions, identifying the domains of the resulting functions, and simplifying them.

Q1:

Simplify the function 𝑛(𝑥)=8𝑥6+𝑥6𝑥6𝑥, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 𝑥 1 4 𝑥 ( 𝑥 6 ) , domain ={0,6}
  • B 𝑛 ( 𝑥 ) = 7 𝑥 + 6 𝑥 ( 𝑥 6 ) , domain ={0,6}
  • C 𝑛 ( 𝑥 ) = 𝑥 1 4 𝑥 ( 𝑥 6 ) , domain ={0,6}
  • D 𝑛 ( 𝑥 ) = 7 𝑥 + 6 𝑥 ( 𝑥 6 ) , domain ={0,6}
  • E 𝑛 ( 𝑥 ) = 𝑥 1 4 𝑥 5 𝑥 6 , domain ={0,6}

Q2:

Simplify the function 𝑛(𝑥)=𝑥7𝑥3𝑥28𝑥77𝑥, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 4 , domain ={4,7}
  • B 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 + 4 , domain ={4}
  • C 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 + 4 , domain ={4,7}
  • D 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 4 , domain ={4}
  • E 𝑛 ( 𝑥 ) = 2 𝑥 + 4 , domain ={4,7}

Q3:

Answer the following questions for the rational expressions 𝑥+33 and 𝑥82𝑥.

Find the sum of 𝑥+33 and 𝑥82𝑥.

  • A 𝑥 + 9 𝑥 2 4 6 𝑥
  • B 𝑥 + 9 𝑥 2 4 6 𝑥
  • C 2 𝑥 + 9 𝑥 2 4 6 𝑥
  • D 2 𝑥 + 3 𝑥 + 5 𝑥
  • E 2 𝑥 + 3 𝑥 5 𝑥

Is the sum of 𝑥+33 and 𝑥82𝑥 a rational expression?

  • Ano
  • Byes

Would this be true for any two rational expressions summed together?

  • Ayes
  • Bno

Q4:

Simplify the function 𝑛(𝑥)=7𝑥𝑥1+3𝑥1𝑥, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 𝑥 ( 7 𝑥 3 ) 𝑥 1 , domain ={1,1}
  • B 𝑛 ( 𝑥 ) = 𝑥 ( 7 𝑥 + 3 ) 𝑥 1 , domain ={1}
  • C 𝑛 ( 𝑥 ) = 7 𝑥 + 3 𝑥 ( 𝑥 1 ) ( 1 𝑥 ) , domain ={1}
  • D 𝑛 ( 𝑥 ) = 𝑥 ( 7 𝑥 3 ) 𝑥 1 , domain ={1}
  • E 𝑛 ( 𝑥 ) = 7 𝑥 + 3 𝑥 ( 𝑥 1 ) ( 1 𝑥 ) , domain ={1,1}

Q5:

Simplify the function 𝑛(𝑥)=𝑥+7𝑥+6+𝑥3𝑥, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 4 𝑥 + 1 3 3 𝑥 , domain ={0,3}
  • B 𝑛 ( 𝑥 ) = 4 𝑥 + 1 3 3 𝑥 , domain ={0}
  • C 𝑛 ( 𝑥 ) = 4 𝑥 + 2 7 3 𝑥 , domain ={0}
  • D 𝑛 ( 𝑥 ) = 4 𝑥 + 2 7 3 𝑥 , domain ={0,3}
  • E 𝑛 ( 𝑥 ) = 2 𝑥 + 1 3 4 𝑥 , domain ={0}

Q6:

Simplify the function 𝑛(𝑥)=3𝑥𝑥+47𝑥𝑥4, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 4 𝑥 ( 𝑥 + 4 ) ( 𝑥 4 ) , domain ={4,4}
  • B 𝑛 ( 𝑥 ) = 4 𝑥 ( 𝑥 + 4 ) ( 𝑥 4 ) , domain ={4,4}
  • C 𝑛 ( 𝑥 ) = 4 𝑥 ( 𝑥 1 0 ) ( 𝑥 + 4 ) ( 𝑥 4 ) , domain ={4,4}
  • D 𝑛 ( 𝑥 ) = 4 𝑥 ( 𝑥 + 1 0 ) ( 𝑥 + 4 ) ( 𝑥 4 ) , domain ={4,4}
  • E 𝑛 ( 𝑥 ) = 4 𝑥 ( 𝑥 + 1 0 ) ( 𝑥 + 4 ) ( 𝑥 4 ) , domain ={4,4}

Q7:

Given that the domain of the function 𝑛(𝑥)=𝑏𝑥+6𝑥+𝑎 is {4,0}, and 𝑛(1)=2, find the values of 𝑎 and 𝑏.

  • A 𝑎 = 4 , 𝑏 = 2
  • B 𝑎 = 4 , 𝑏 = 2
  • C 𝑎 = 4 , 𝑏 = 0
  • D 𝑎 = 4 , 𝑏 = 0
  • E 𝑎 = 4 , 𝑏 = 4

Q8:

Simplify the function 𝑛(𝑥)=8𝑥+7𝑥14𝑥+45+3𝑥24𝑥17𝑥+72, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 1 1 𝑥 1 7 2 𝑥 3 1 𝑥 + 1 1 7 , domain ={5,8,9}
  • B 𝑛 ( 𝑥 ) = 2 ( 4 𝑥 + 5 ) ( 𝑥 9 ) ( 𝑥 5 ) , domain ={5,8,9}
  • C 𝑛 ( 𝑥 ) = 1 1 𝑥 8 ( 𝑥 9 ) ( 𝑥 5 ) , domain ={5,9}
  • D 𝑛 ( 𝑥 ) = 2 ( 4 𝑥 + 5 ) ( 𝑥 9 ) ( 𝑥 5 ) , domain ={5,9}
  • E 𝑛 ( 𝑥 ) = 1 1 𝑥 8 ( 𝑥 9 ) ( 𝑥 5 ) , domain ={5,8,9}

Q9:

Simplify the function 𝑛(𝑥)=𝑥+𝑥+1𝑥181𝑥𝑥10𝑥+9, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 𝑥 + 1 0 𝑥 1 , domain ={1,9}
  • B 𝑛 ( 𝑥 ) = 𝑥 8 𝑥 1 , domain ={1,9}
  • C 𝑛 ( 𝑥 ) = 𝑥 + 1 0 𝑥 1 , domain ={1}
  • D 𝑛 ( 𝑥 ) = 𝑥 8 𝑥 1 , domain ={1,9}
  • E 𝑛 ( 𝑥 ) = 𝑥 8 𝑥 1 , domain ={1}

Q10:

Answer the following questions for the rational expressions 5𝑥23𝑥 and 3𝑥2𝑥2𝑥+8.

Subtract 5𝑥23𝑥 from 3𝑥2𝑥2𝑥+8.

  • A 𝑥 1 6 𝑥 + 3 6 𝑥 𝑥 ( 𝑥 + 4 )
  • B 9 𝑥 1 6 𝑥 3 6 𝑥 + 1 6 6 𝑥 ( 𝑥 + 4 )
  • C 𝑥 + 2 𝑥 3 6 𝑥 + 1 6 6 𝑥 ( 𝑥 + 4 )
  • D 9 𝑥 2 𝑥 3 6 𝑥 + 1 6 6 𝑥 ( 𝑥 + 4 )
  • E 𝑥 3 𝑥 + 3 6 𝑥 𝑥 ( 𝑥 + 4 )

Is the difference between 3𝑥2𝑥2𝑥+8 and 5𝑥23𝑥 a rational expression?

  • Ayes
  • Bno

Is the result of this subtraction a rational expression?

  • Ano
  • Byes

Q11:

Simplify the function 𝑛(𝑥)=9𝑥+6+9𝑥6, and determine its domain in .

  • A 𝑛 ( 𝑥 ) = 1 8 ( 𝑥 6 ) ( 𝑥 + 6 ) , domain ={6,6}
  • B 𝑛 ( 𝑥 ) = 1 8 𝑥 ( 𝑥 6 ) ( 𝑥 + 6 ) , domain ={6}
  • C 𝑛 ( 𝑥 ) = 1 8 𝑥 ( 𝑥 6 ) ( 𝑥 + 6 ) , domain ={6,6}
  • D 𝑛 ( 𝑥 ) = 9 𝑥 , domain ={6,6}

Q12:

Simplify the function 𝑛(𝑥)=5𝑥𝑥4𝑥+4𝑥16, and determine its domain.

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

Q13:

Simplify the function 𝑛(𝑥)=𝑥+3𝑥493𝑥+21𝑥3𝑥6𝑥105𝑥, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 2 ( 1 1 𝑥 + 3 2 ) ( 𝑥 7 ) ( 𝑥 + 7 ) ( 𝑥 + 5 ) , domain ={7,5,7}
  • B 𝑛 ( 𝑥 ) = 2 ( 3 𝑥 + 1 7 ) ( 𝑥 7 ) ( 𝑥 + 7 ) ( 𝑥 + 5 ) , domain ={7,5,7}
  • C 𝑛 ( 𝑥 ) = 2 ( 3 𝑥 + 1 7 ) ( 𝑥 7 ) ( 𝑥 + 7 ) ( 𝑥 + 5 ) , domain ={7,5,0,7}
  • D 𝑛 ( 𝑥 ) = 3 𝑥 2 1 𝑥 + 𝑥 + 3 ( 𝑥 7 ) ( 𝑥 + 7 ) ( 𝑥 + 5 ) , domain ={7,5,0,7}
  • E 𝑛 ( 𝑥 ) = 2 ( 1 1 𝑥 + 3 2 ) ( 𝑥 7 ) ( 𝑥 + 7 ) ( 𝑥 + 5 ) , domain ={7,5,0,7}

Q14:

Simplify the function 𝑛(𝑥)=𝑥+13𝑥𝑥3𝑥8, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 1 8 , domain =
  • B 𝑛 ( 𝑥 ) = 5 𝑥 + 8 3 𝑥 ( 3 𝑥 8 ) , domain =0,83
  • C 𝑛 ( 𝑥 ) = 1 8 , domain =0,83
  • D 𝑛 ( 𝑥 ) = 5 𝑥 + 8 3 𝑥 ( 3 𝑥 8 ) , domain =0,83

Q15:

Simplify the function 𝑛(𝑥)=(𝑥8)𝑥𝑥+8, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 𝑥 + 𝑥 8 𝑥 + 8 , domain ={8}
  • B 𝑛 ( 𝑥 ) = 6 4 𝑥 + 8 , domain ={8}
  • C 𝑛 ( 𝑥 ) = 6 4 𝑥 + 8 , domain ={8}
  • D 𝑛 ( 𝑥 ) = 6 4 𝑥 + 8 , domain ={8}
  • E 𝑛 ( 𝑥 ) = 6 4 𝑥 + 8 , domain ={8}

Q16:

Simplify the function 𝑛(𝑥)=64𝑥1+99𝑥18𝑥, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 4 𝑥 1 𝑥 ( 2 𝑥 + 1 ) ( 2 𝑥 1 ) , domain =12,0,12
  • B 𝑛 ( 𝑥 ) = 1 5 1 4 𝑥 + 9 𝑥 1 , domain =12,0,12
  • C 𝑛 ( 𝑥 ) = 5 𝑥 ( 2 𝑥 + 1 ) ( 2 𝑥 1 ) , domain =12,0,12
  • D 𝑛 ( 𝑥 ) = 4 𝑥 1 𝑥 ( 2 𝑥 + 1 ) ( 2 𝑥 1 ) , domain =12,12

Q17:

Simplify the function 𝑛(𝑥)=5𝑥+8+7𝑥+464𝑥, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 1 2 ( 𝑥 3 ) ( 𝑥 8 ) ( 𝑥 + 8 ) , domain ={8,8}
  • B 𝑛 ( 𝑥 ) = 4 ( 3 𝑥 + 1 1 ) ( 𝑥 8 ) ( 𝑥 + 8 ) , domain ={8,8}
  • C 𝑛 ( 𝑥 ) = 1 2 ( 𝑥 3 ) ( 𝑥 8 ) ( 𝑥 + 8 ) , domain ={8}
  • D 𝑛 ( 𝑥 ) = 4 ( 3 𝑥 + 1 1 ) ( 𝑥 8 ) ( 𝑥 + 8 ) , domain ={8}
  • E 𝑛 ( 𝑥 ) = 1 2 ( 𝑥 + 3 ) ( 𝑥 8 ) ( 𝑥 + 8 ) , domain ={8,8}

Q18:

Simplify the function 𝑛(𝑥)=𝑥68𝑥57𝑥+54+𝑥581𝑥818𝑥, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 4 ( 8 𝑥 9 ) ( 𝑥 9 ) , domain =98,9
  • B 𝑛 ( 𝑥 ) = 1 4 ( 8 𝑥 + 9 ) ( 𝑥 + 9 ) , domain =98,6,9
  • C 𝑛 ( 𝑥 ) = 4 ( 8 𝑥 9 ) ( 𝑥 9 ) , domain =98,6,9
  • D 𝑛 ( 𝑥 ) = 1 4 ( 8 𝑥 9 ) ( 𝑥 9 ) , domain =98,9
  • E 𝑛 ( 𝑥 ) = 1 4 ( 8 𝑥 9 ) ( 𝑥 9 ) , domain =98,6,9

Q19:

Simplify the function 𝑛(𝑥)=𝑥4𝑥32𝑥16𝑥+64+𝑥8𝑥+12𝑥10𝑥+16, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 2 ( 𝑥 + 1 ) 𝑥 8 , domain ={8}
  • B 𝑛 ( 𝑥 ) = 2 ( 𝑥 1 ) 𝑥 8 , domain ={2,8}
  • C 𝑛 ( 𝑥 ) = 2 ( 𝑥 1 ) 𝑥 8 , domain ={8}
  • D 𝑛 ( 𝑥 ) = 2 ( 𝑥 + 1 ) 𝑥 8 , domain ={2,8}
  • E 𝑛 ( 𝑥 ) = 2 ( 𝑥 1 ) 𝑥 + 8 , domain ={2,8}

Q20:

Simplify the function 𝑛(𝑥)=𝑥76𝑥35𝑥49+𝑥+214+19𝑥+6𝑥, and determine its domain.

  • A 𝑛 ( 𝑥 ) = 2 6 𝑥 + 7 , domain =76
  • B 𝑛 ( 𝑥 ) = 2 6 𝑥 + 7 , domain =7,2,76
  • C 𝑛 ( 𝑥 ) = 2 𝑥 5 1 2 𝑥 1 6 𝑥 3 5 , domain =7,2,76
  • D 𝑛 ( 𝑥 ) = 2 6 𝑥 7 , domain =7,2,76
  • E 𝑛 ( 𝑥 ) = 2 6 𝑥 7 , domain =76

Q21:

Given that the domain of the function 𝑛(𝑥)=2𝑥(𝑥𝑎)(𝑥+6)+5𝑥+5(𝑥𝑎)(𝑥+3) is {6,3,2}, what is the value of 𝑎?

  • A { 6 , 3 , 2 }
  • B { 2 }
  • C { 2 }
  • D { 2 }
  • E { 2 }

Q22:

Simplify the function 𝑛(𝑥)=4𝑥8𝑥+2𝑥8+2𝑥+4𝑥+4×𝑥+27𝑥3𝑥+9, and find its domain.

  • A 𝑛 ( 𝑥 ) = 2 ( 𝑥 + 3 ) , domain ={4,2}
  • B 𝑛 ( 𝑥 ) = 2 ( 𝑥 + 3 ) , domain ={4,3,2}
  • C 𝑛 ( 𝑥 ) = 2 ( 𝑥 3 ) , domain ={4,3,2}
  • D 𝑛 ( 𝑥 ) = 2 𝑥 + 3 , domain ={4,2}
  • E 𝑛 ( 𝑥 ) = 2 ( 𝑥 3 ) , domain ={4,2}

Q23:

Simplify 5𝑥23𝑥7𝑥29𝑥.

  • A 2 3 𝑥 9 𝑥
  • B 5 𝑥 6 𝑥 + 7 𝑥 9 𝑥
  • C 1 5 𝑥 6 𝑥 7 𝑥 + 2 9 𝑥
  • D 2 9 𝑥
  • E 1 5 𝑥 6 𝑥 7 𝑥 + 2 9 𝑥

Q24:

Simplify 3𝑥+27𝑥+3𝑥2𝑥.

  • A 3 𝑥 + 3 𝑥 + 2 7 𝑥 ( 2 𝑥 )
  • B 2 1 𝑥 3 𝑥 + 4 𝑥 + 4 7 𝑥 ( 2 𝑥 )
  • C 3 𝑥 3 𝑥 + 2 𝑥 ( 2 𝑥 )
  • D 2 1 𝑥 3 𝑥 + 4 𝑥 + 4 7 ( 2 𝑥 )
  • E 3 𝑥 + 3 𝑥 + 2 6 𝑥 + 2

Q25:

Simplify 3𝑥+2𝑥+4𝑥+4+3𝑥𝑥4.

  • A 3 𝑥 + 1 5 𝑥 + 1 4 𝑥 1 2 𝑥 8 ( 𝑥 + 4 𝑥 + 4 ) ( 𝑥 4 )
  • B 3 𝑥 + 9 𝑥 4 𝑥 4 ( 𝑥 2 ) ( 𝑥 + 2 )
  • C 3 𝑥 + 9 𝑥 4 𝑥 4 ( 𝑥 2 ) ( 𝑥 + 2 )
  • D 3 𝑥 + 9 𝑥 4 𝑥 4 ( 𝑥 2 ) ( 𝑥 + 2 )
  • E 3 𝑥 + 1 5 𝑥 + 1 4 𝑥 1 2 𝑥 8 ( 𝑥 + 2 ) ( 𝑥 2 )

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