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.

  • A𝑛(𝑥)=𝑥14𝑥(𝑥6), domain ={0,6}
  • B𝑛(𝑥)=7𝑥+6𝑥(𝑥6), domain ={0,6}
  • C𝑛(𝑥)=𝑥14𝑥(𝑥6), domain ={0,6}
  • D𝑛(𝑥)=7𝑥+6𝑥(𝑥6), domain ={0,6}
  • E𝑛(𝑥)=𝑥14𝑥5𝑥6, domain ={0,6}

Q2:

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

  • A𝑛(𝑥)=𝑥+5𝑥4, domain ={4,7}
  • B𝑛(𝑥)=𝑥+5𝑥+4, domain ={4}
  • C𝑛(𝑥)=𝑥+5𝑥+4, domain ={4,7}
  • D𝑛(𝑥)=𝑥+5𝑥4, domain ={4}
  • E𝑛(𝑥)=2𝑥+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.

  • A𝑥+9𝑥246𝑥
  • B𝑥+9𝑥246𝑥
  • C2𝑥+9𝑥246𝑥
  • D2𝑥+3𝑥+5𝑥
  • E2𝑥+3𝑥5𝑥

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.

  • A𝑛(𝑥)=𝑥(7𝑥3)𝑥1, domain ={1,1}
  • B𝑛(𝑥)=𝑥(7𝑥+3)𝑥1, domain ={1}
  • C𝑛(𝑥)=7𝑥+3𝑥(𝑥1)(1𝑥), domain ={1}
  • D𝑛(𝑥)=𝑥(7𝑥3)𝑥1, domain ={1}
  • E𝑛(𝑥)=7𝑥+3𝑥(𝑥1)(1𝑥), domain ={1,1}

Q5:

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

  • A𝑛(𝑥)=4𝑥+133𝑥, domain ={0,3}
  • B𝑛(𝑥)=4𝑥+133𝑥, domain ={0}
  • C𝑛(𝑥)=4𝑥+273𝑥, domain ={0}
  • D𝑛(𝑥)=4𝑥+273𝑥, domain ={0,3}
  • E𝑛(𝑥)=2𝑥+134𝑥, domain ={0}

Q6:

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

  • A𝑛(𝑥)=4𝑥(𝑥+4)(𝑥4), domain ={4,4}
  • B𝑛(𝑥)=4𝑥(𝑥+4)(𝑥4), domain ={4,4}
  • C𝑛(𝑥)=4𝑥(𝑥10)(𝑥+4)(𝑥4), domain ={4,4}
  • D𝑛(𝑥)=4𝑥(𝑥+10)(𝑥+4)(𝑥4), domain ={4,4}
  • E𝑛(𝑥)=4𝑥(𝑥+10)(𝑥+4)(𝑥4), 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.

  • A𝑎=4, 𝑏=2
  • B𝑎=4, 𝑏=2
  • C𝑎=4, 𝑏=0
  • D𝑎=4, 𝑏=0
  • E𝑎=4, 𝑏=4

Q8:

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

  • A𝑛(𝑥)=11𝑥172𝑥31𝑥+117, domain ={5,8,9}
  • B𝑛(𝑥)=2(4𝑥+5)(𝑥9)(𝑥5), domain ={5,8,9}
  • C𝑛(𝑥)=11𝑥8(𝑥9)(𝑥5), domain ={5,9}
  • D𝑛(𝑥)=2(4𝑥+5)(𝑥9)(𝑥5), domain ={5,9}
  • E𝑛(𝑥)=11𝑥8(𝑥9)(𝑥5), domain ={5,8,9}

Q9:

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

  • A𝑛(𝑥)=𝑥+10𝑥1, domain ={1,9}
  • B𝑛(𝑥)=𝑥8𝑥1, domain ={1,9}
  • C𝑛(𝑥)=𝑥+10𝑥1, domain ={1}
  • D𝑛(𝑥)=𝑥8𝑥1, domain ={1,9}
  • E𝑛(𝑥)=𝑥8𝑥1, domain ={1}

Q10:

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

Subtract 5𝑥23𝑥 from 3𝑥2𝑥2𝑥+8.

  • A𝑥16𝑥+36𝑥𝑥(𝑥+4)
  • B9𝑥16𝑥36𝑥+166𝑥(𝑥+4)
  • C𝑥+2𝑥36𝑥+166𝑥(𝑥+4)
  • D9𝑥2𝑥36𝑥+166𝑥(𝑥+4)
  • E𝑥3𝑥+36𝑥𝑥(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 .

  • A𝑛(𝑥)=18(𝑥6)(𝑥+6), domain ={6,6}
  • B𝑛(𝑥)=18𝑥(𝑥6)(𝑥+6), domain ={6}
  • C𝑛(𝑥)=18𝑥(𝑥6)(𝑥+6), domain ={6,6}
  • D𝑛(𝑥)=9𝑥, domain ={6,6}

Q12:

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

  • A𝑛(𝑥)=5𝑥+1𝑥4, domain ={4}
  • B𝑛(𝑥)=4(𝑥1)(𝑥4)(𝑥16), domain ={4,4}
  • C𝑛(𝑥)=5𝑥1𝑥4, domain ={4}
  • D𝑛(𝑥)=5𝑥+1𝑥4, domain ={4,4}
  • E𝑛(𝑥)=5𝑥1𝑥4, domain ={4,4}

Q13:

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

  • A𝑛(𝑥)=2(11𝑥+32)(𝑥7)(𝑥+7)(𝑥+5), domain ={7,5,7}
  • B𝑛(𝑥)=2(3𝑥+17)(𝑥7)(𝑥+7)(𝑥+5), domain ={7,5,7}
  • C𝑛(𝑥)=2(3𝑥+17)(𝑥7)(𝑥+7)(𝑥+5), domain ={7,5,0,7}
  • D𝑛(𝑥)=3𝑥21𝑥+𝑥+3(𝑥7)(𝑥+7)(𝑥+5), domain ={7,5,0,7}
  • E𝑛(𝑥)=2(11𝑥+32)(𝑥7)(𝑥+7)(𝑥+5), domain ={7,5,0,7}

Q14:

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

  • A𝑛(𝑥)=18, domain =
  • B𝑛(𝑥)=5𝑥+83𝑥(3𝑥8), domain =0,83
  • C𝑛(𝑥)=18, domain =0,83
  • D𝑛(𝑥)=5𝑥+83𝑥(3𝑥8), domain =0,83

Q15:

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

  • A𝑛(𝑥)=𝑥+𝑥8𝑥+8, domain ={8}
  • B𝑛(𝑥)=64𝑥+8, domain ={8}
  • C𝑛(𝑥)=64𝑥+8, domain ={8}
  • D𝑛(𝑥)=64𝑥+8, domain ={8}
  • E𝑛(𝑥)=64𝑥+8, domain ={8}

Q16:

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

  • A𝑛(𝑥)=4𝑥1𝑥(2𝑥+1)(2𝑥1), domain =12,0,12
  • B𝑛(𝑥)=1514𝑥+9𝑥1, domain =12,0,12
  • C𝑛(𝑥)=5𝑥(2𝑥+1)(2𝑥1), domain =12,0,12
  • D𝑛(𝑥)=4𝑥1𝑥(2𝑥+1)(2𝑥1), domain =12,12

Q17:

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

  • A𝑛(𝑥)=12(𝑥3)(𝑥8)(𝑥+8), domain ={8,8}
  • B𝑛(𝑥)=4(3𝑥+11)(𝑥8)(𝑥+8), domain ={8,8}
  • C𝑛(𝑥)=12(𝑥3)(𝑥8)(𝑥+8), domain ={8}
  • D𝑛(𝑥)=4(3𝑥+11)(𝑥8)(𝑥+8), domain ={8}
  • E𝑛(𝑥)=12(𝑥+3)(𝑥8)(𝑥+8), domain ={8,8}

Q18:

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

  • A𝑛(𝑥)=4(8𝑥9)(𝑥9), domain =98,9
  • B𝑛(𝑥)=14(8𝑥+9)(𝑥+9), domain =98,6,9
  • C𝑛(𝑥)=4(8𝑥9)(𝑥9), domain =98,6,9
  • D𝑛(𝑥)=14(8𝑥9)(𝑥9), domain =98,9
  • E𝑛(𝑥)=14(8𝑥9)(𝑥9), domain =98,6,9

Q19:

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

  • A𝑛(𝑥)=2(𝑥+1)𝑥8, domain ={8}
  • B𝑛(𝑥)=2(𝑥1)𝑥8, domain ={2,8}
  • C𝑛(𝑥)=2(𝑥1)𝑥8, domain ={8}
  • D𝑛(𝑥)=2(𝑥+1)𝑥8, domain ={2,8}
  • E𝑛(𝑥)=2(𝑥1)𝑥+8, domain ={2,8}

Q20:

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

  • A𝑛(𝑥)=26𝑥+7, domain =7,2,76
  • B𝑛(𝑥)=26𝑥+7, domain =76
  • C𝑛(𝑥)=26𝑥7, domain =76
  • D𝑛(𝑥)=2𝑥512𝑥16𝑥35, domain =7,2,76
  • E𝑛(𝑥)=26𝑥7, domain =7,2,76

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.

  • A𝑛(𝑥)=2(𝑥+3), domain ={4,2}
  • B𝑛(𝑥)=2(𝑥+3), domain ={4,3,2}
  • C𝑛(𝑥)=2(𝑥3), domain ={4,3,2}
  • D𝑛(𝑥)=2𝑥+3, domain ={4,2}
  • E𝑛(𝑥)=2(𝑥3), domain ={4,2}

Q23:

Simplify 5𝑥23𝑥7𝑥29𝑥.

  • A23𝑥9𝑥
  • B5𝑥6𝑥+7𝑥9𝑥
  • C15𝑥6𝑥7𝑥+29𝑥
  • D29𝑥
  • E15𝑥6𝑥7𝑥+29𝑥

Q24:

Simplify 3𝑥+27𝑥+3𝑥2𝑥.

  • A3𝑥+3𝑥+27𝑥(2𝑥)
  • B21𝑥3𝑥+4𝑥+47𝑥(2𝑥)
  • C3𝑥3𝑥+2𝑥(2𝑥)
  • D21𝑥3𝑥+4𝑥+47(2𝑥)
  • E3𝑥+3𝑥+26𝑥+2

Q25:

Simplify 3𝑥+2𝑥+4𝑥+4+3𝑥𝑥4.

  • A3𝑥+15𝑥+14𝑥12𝑥8(𝑥+4𝑥+4)(𝑥4)
  • B3𝑥+9𝑥4𝑥4(𝑥2)(𝑥+2)
  • C3𝑥+9𝑥4𝑥4(𝑥2)(𝑥+2)
  • D3𝑥+9𝑥4𝑥4(𝑥2)(𝑥+2)
  • E3𝑥+15𝑥+14𝑥12𝑥8(𝑥+2)(𝑥2)

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