Worksheet: Domain and Range of a Rational Function

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In this worksheet, we will practice finding the domain and range of a rational function either from its graph or its defining rule.

Q1:

Find the value of 𝑐 given 𝑛 ( 𝑥 ) = 1 4 2 5 𝑥 + 6 0 𝑥 + 3 6 where 𝑛 ( 𝑐 ) is undefined.

  • A − 5 6
  • B 6 5
  • C 5 6
  • D − 6 5
  • E14

Q2:

Determine the domain and the range of the function 𝑓 ( 𝑥 ) = 1 𝑥 − 2 .

  • A The domain is { 0 } , and the range is ℝ .
  • B The domain is ℝ , and the range is ℝ .
  • C The domain is ℝ , and the range is { 2 } .
  • D The domain is ℝ − { 2 } , and the range is ℝ − { 0 } .

Q3:

Simplify the function 𝑓 ( 𝑥 ) = 𝑥 + 7 𝑥 − 8 𝑥 𝑥 − 6 5 𝑥 + 6 4 , and find its domain.

  • A 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 − 1 ) ( 𝑥 + 8 ) , domain = ℝ − { 1 , − 1 , − 8 , 8 }
  • B 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 + 1 ) ( 𝑥 − 8 ) , domain = ℝ − { − 1 , 8 }
  • C 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 − 1 ) ( 𝑥 + 8 ) , domain = ℝ − { 1 , − 8 }
  • D 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 + 1 ) ( 𝑥 − 8 ) , domain = ℝ − { 1 , − 1 , − 8 , 8 }
  • E 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 + 7 𝑥 − 8 ) ( 𝑥 − 1 ) ( 𝑥 − 6 4 ) , domain = ℝ − { 1 , − 1 , − 8 , 8 }

Q4:

Find the domain of the function 𝑓 ( 𝑥 ) = 𝑥 + 4 ( 𝑥 − 8 ) .

  • A ℝ − { − 8 , 8 }
  • B ℝ − { − 4 , 8 }
  • C ℝ − { − 4 }
  • D ℝ − { 8 }

Q5:

The domain of an algebraic fractional function is the set of all real numbers except the .

  • A zeros of the denominator of the function
  • B zeros of the function

Q6:

Determine the domain and the range of the function 𝑓 ( 𝑥 ) = 𝑥 − 1 6 𝑥 + 4 .

  • A The domain is { − 4 } and the range is ℝ .
  • B The domain is ℝ − { − 8 } and the range is ℝ − { − 4 } .
  • C The domain is ℝ and the range is { − 4 } .
  • D The domain is ℝ − { − 4 } and the range is ℝ − { − 8 } .
  • E The domain is { − 8 , − 4 } and the range is ℝ .

Q7:

Define a function on real numbers by 𝑓 ( 𝑥 ) = 2 𝑥 + 3 4 𝑥 + 5 .

What is the domain of this function?

  • Aall real numbers
  • Ball real numbers except 5 4
  • Call real numbers except 3 2
  • Dall real numbers except − 5 4
  • Eall real numbers except − 3 2

Find the one value that 𝑓 ( 𝑥 ) cannot take.

  • A 1 2
  • B − 3 2
  • C 5 4
  • D − 5 4
  • E − 1 2

What is the range of this function?

  • Aall real numbers except − 1 2
  • Ball real numbers
  • Call real numbers except 1 2
  • Dall real numbers except − 5 4
  • Eall real numbers except − 3 2

Q8:

Given that the domain of the function 𝑛 ( 𝑥 ) = 3 6 𝑥 + 2 0 𝑥 + 𝑎 is ℝ − { − 2 , 0 } , evaluate 𝑛 ( 3 ) .

Q9:

Determine the domain and the range of the function 𝑓 ( 𝑥 ) = 1 | 𝑥 − 2 | .

  • A The domain is { 2 } and the range is ℝ .
  • B The domain is ] 0 , ∞ [ and the range is ℝ − { 2 } .
  • C The domain is ℝ and the range is { 2 } .
  • D The domain is ℝ − { 2 } and the range is ] 0 , ∞ [ .

Q10:

Identify the domain of 𝑛 ( 𝑥 ) = 9 𝑥 + 8 3 𝑥 + 2 .

  • A ℝ − − 3 2
  • B ℝ − 2 3
  • C ℝ − 3 2
  • D ℝ − − 2 3
  • E ℝ − − 8 9 , − 2 3

Q11:

Determine the domain of the function 𝑛 ( 𝑥 ) = 4 𝑥 + 3 𝑥 6 𝑥 + 7 𝑥 .

  • A ℝ − − 3 4
  • B ℝ − 0 , − 3 4 , − 7 6
  • C ℝ − − 7 6
  • D ℝ − 0 , − 7 6
  • E ℝ − { 0 }

Q12:

For which values of 𝑥 is the function 𝑛 ( 𝑥 ) = 𝑥 − 2 5 𝑥 − 1 2 𝑥 + 3 2 not defined?

  • A { − 5 , 5 }
  • B ℝ − { 4 , 8 }
  • C ℝ − { − 5 , 5 }
  • D { 4 , 8 }
  • E { − 8 , − 4 }

Q13:

If the common domain of the two functions 𝑛 ( 𝑥 ) = 3 𝑥 + 3 and 𝑛 ( 𝑥 ) = 3 𝑥 𝑥 − 𝑚 is ℝ − { − 3 , 4 } , what is the value of 𝑚 ?

Q14:

The function 𝑓 ( 𝑥 ) = 𝑥 − 2 𝑥 − 1 has an additive inverse if the domain is .

  • A ℝ − { 2 }
  • B ℝ − { − 1 }
  • C ℝ − { − 2 }
  • D ℝ − { 1 }
  • E ℝ − { 1 , 2 }

Q15:

If the function 𝑛 ( 𝑥 ) = 𝑥 + 3 𝑥 − 9 , what is the domain of its multiplicative inverse?

  • A ℝ − { − 3 }
  • B ℝ − { − 9 , 3 }
  • C ℝ − { 9 }
  • D ℝ − { − 3 , 9 }

Q16:

Determine the domain and the range of the function .

  • A The domain is , the range is .
  • B The domain is , the range is .
  • C The domain is , the range is .
  • D The domain is , the range is .

Q17:

Determine the domain and the range of 𝑓 ( 𝑥 ) = 𝑥 − 4 𝑥 − 2 .

  • A The domain is { 2 } and the range is ℝ .
  • B The domain is ℝ − { 4 } and the range is ℝ − { 2 } .
  • C The domain is ℝ and the range is { 2 } .
  • D The domain is ℝ − { 2 } and the range is ℝ − { 4 } .
  • E The domain is { 2 , 4 } and the range is ℝ .

Q18:

Determine the domain of the function 𝑓 ( 𝑥 ) = 𝑥 + 3 | 𝑥 − 1 | + 7 .

  • A ℝ − { 1 }
  • B ℝ − { − 8 , 8 }
  • C ℝ − { 8 }
  • D ℝ

Q19:

Given the function 𝑛 ( 𝑥 ) = 𝑥 + 8 𝑥 ( 𝑥 + 8 ) ( 𝑥 + 5 ) , find the multiplicative inverse of 𝑛 in its simplest form and state its domain.

  • A 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 , domain = ℝ − { 0 }
  • B 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 , domain = ℝ − { − 8 }
  • C 𝑛 ( 𝑥 ) = 𝑥 𝑥 + 5 , domain = ℝ − { − 8 }
  • D 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 , domain = ℝ − { − 8 , 0 }
  • E 𝑛 ( 𝑥 ) = 𝑥 𝑥 + 5 , domain = ℝ − { − 8 , 0 }

Q20:

Determine the domain of the function 𝑓 ( 𝑥 ) = | 𝑥 − 1 | | 𝑥 | − 8 .

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

Q21:

Determine the domain and the range of the function 𝑓 ( 𝑥 ) = 𝑥 + 9 𝑥 + 2 0 𝑥 + 5 .

  • A The domain is { − 5 } and the range is ℝ .
  • B The domain is ℝ − { − 1 } and the range is ℝ − { − 5 } .
  • C The domain is ℝ and the range is { − 5 } .
  • D The domain is ℝ − { − 5 } and the range is ℝ − { − 1 } .

Q22:

Determine the domain and the range of the function 𝑓 ( 𝑥 ) = − 1 0 𝑥 + 6 4 0 𝑥 − 6 4 .

  • A The domain is { − 1 0 } , and the range is ℝ .
  • B The domain is { − 1 0 } , and the range is ℝ − { − 8 , 8 } .
  • C The domain is ℝ , and the range is { − 1 0 } .
  • D The domain is ℝ − { − 8 , 8 } , and the range is { − 1 0 } .
  • E The domain is { − 8 , 8 } , and the range is ℝ .

Q23:

Determine the domain and the range of the function 𝑓 ( 𝑥 ) = − 5 𝑥 − 1 5 𝑥 + 3 .

  • A The domain is ℝ and the range is { − 5 } .
  • B The domain is { − 5 } and the range is ℝ − { − 3 } .
  • C The domain is { − 5 } and the range is ℝ .
  • D The domain is ℝ − { − 3 } and the range is { − 5 } .

Q24:

Given that the domain of the function 𝑛 ( 𝑥 ) = 8 𝑥 − 9 1 is the same as the domain of the function 𝑛 ( 𝑥 ) = 𝑥 − 5 𝑥 + 𝑙 2 , what is the value of 𝑙 ?

Q25:

Determine the domain and the range of the function 𝑓 ( 𝑥 ) = 𝑥 + 2 𝑥 − 8 𝑥 + 4 .

  • A The domain is { − 4 } and the range is ℝ .
  • B The domain is ℝ − { − 6 } and the range is ℝ − { − 4 } .
  • C The domain is ℝ and the range is { − 4 } .
  • D The domain is ℝ − { − 4 } and the range is ℝ − { − 6 } .
  • E The domain is { − 6 , − 4 } and the range is ℝ .

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