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Worksheet: Domain and Range of a Rational Function

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 domain of the function 𝑓 ( π‘₯ ) = π‘₯ + 4 ( π‘₯ βˆ’ 8 ) 2 .

  • A ℝ βˆ’ { βˆ’ 8 , 8 }
  • B ℝ βˆ’ { βˆ’ 4 , 8 }
  • C ℝ βˆ’ { βˆ’ 4 }
  • D ℝ βˆ’ { 8 }

Q2:

Find the domain of the function 𝑓 ( π‘₯ ) = π‘₯ + 1 ( π‘₯ βˆ’ 8 ) 2 .

  • A ℝ βˆ’ { βˆ’ 8 , 8 }
  • B ℝ βˆ’ { βˆ’ 1 , 8 }
  • C ℝ βˆ’ { βˆ’ 1 }
  • D ℝ βˆ’ { 8 }

Q3:

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

  • A βˆ’ 5 6
  • B 6 5
  • C 5 6
  • D βˆ’ 6 5
  • E14

Q4:

Determine the domain and the range of the function 𝑓 ( π‘₯ ) = 1 π‘₯ βˆ’ 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 βˆ’ { 2 } , and the range is ℝ βˆ’ { 0 } .

Q5:

If the common domain of the two functions 𝑛 ( π‘₯ ) = 3 π‘₯ + 3 1 and 𝑛 ( π‘₯ ) = 3 π‘₯ π‘₯ βˆ’ π‘š 2 is ℝ βˆ’ { βˆ’ 3 , 4 } , what is the value of π‘š ?

Q6:

The function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 2 π‘₯ βˆ’ 1 has an additive inverse if the domain is .

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

Q7:

If the function 𝑛 ( π‘₯ ) = π‘₯ + 3 π‘₯ βˆ’ 9 , what is the domain of its multiplicative inverse?

  • A ℝ βˆ’ { βˆ’ 3 }
  • B ℝ βˆ’ { βˆ’ 9 , 3 }
  • C ℝ βˆ’ { 9 }
  • D ℝ βˆ’ { βˆ’ 3 , 9 }

Q8:

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 .

Q9:

Determine the domain and the range of the function 𝑓 ( π‘₯ ) = π‘₯ + 2 π‘₯ βˆ’ 8 π‘₯ + 4 2 .

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

Q10:

Simplify the function 𝑓 ( π‘₯ ) = π‘₯ + 7 π‘₯ βˆ’ 8 π‘₯ π‘₯ βˆ’ 6 5 π‘₯ + 6 4 3 2 4 2 , and find its domain.

  • A 𝑓 ( π‘₯ ) = π‘₯ ( π‘₯ βˆ’ 1 ) ( π‘₯ + 8 ) , domain = ℝ βˆ’ { 1 , βˆ’ 1 , βˆ’ 8 , 8 }
  • B 𝑓 ( π‘₯ ) = π‘₯ ( π‘₯ + 1 ) ( π‘₯ βˆ’ 8 ) , domain = ℝ βˆ’ { βˆ’ 1 , 8 }
  • C 𝑓 ( π‘₯ ) = π‘₯ ( π‘₯ βˆ’ 1 ) ( π‘₯ + 8 ) , domain = ℝ βˆ’ { 1 , βˆ’ 8 }
  • D 𝑓 ( π‘₯ ) = π‘₯ ( π‘₯ + 1 ) ( π‘₯ βˆ’ 8 ) , domain = ℝ βˆ’ { 1 , βˆ’ 1 , βˆ’ 8 , 8 }
  • E 𝑓 ( π‘₯ ) = π‘₯ ( π‘₯ + 7 π‘₯ βˆ’ 8 ) ( π‘₯ βˆ’ 1 ) ( π‘₯ βˆ’ 6 4 ) 2 2 2 , domain = ℝ βˆ’ { 1 , βˆ’ 1 , βˆ’ 8 , 8 }

Q11:

Determine the domain and the range of the function .

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

Q12:

Identify the domain of .

  • A
  • B
  • C
  • D
  • E

Q13:

Determine the domain and the range of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 4 π‘₯ βˆ’ 2 2 .

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

Q14:

Determine the domain and the range of the function 𝑓 ( π‘₯ ) = 1 π‘₯ + 5 βˆ’ 1 in ℝ .

  • A The domain is { βˆ’ 5 } , and the range is ℝ .
  • B The domain is ℝ , and the range is ℝ .
  • C The domain is ℝ , and the range is { βˆ’ 5 } .
  • D The domain is ℝ βˆ’ { βˆ’ 5 } , and the range is ℝ βˆ’ { βˆ’ 1 } .
  • E The domain is ℝ βˆ’ { 0 } , and the range is ℝ βˆ’ { 0 } .

Q15:

Determine the domain of the function 𝑓 ( π‘₯ ) = π‘₯ + 3 | π‘₯ βˆ’ 1 | + 7 2 .

  • A ℝ βˆ’ { 1 }
  • B ℝ βˆ’ { βˆ’ 8 , 8 }
  • C ℝ βˆ’ { 8 }
  • D ℝ

Q16:

For which values of π‘₯ is the function 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 2 5 π‘₯ βˆ’ 1 2 π‘₯ + 3 2 2 2 not defined?

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

Q17:

For which values of π‘₯ is the function 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 9 π‘₯ + 1 1 π‘₯ + 2 4 2 2 not defined?

  • A { βˆ’ 3 , 3 }
  • B ℝ βˆ’ { βˆ’ 8 , βˆ’ 3 }
  • C ℝ βˆ’ { βˆ’ 3 , 3 }
  • D { βˆ’ 8 , βˆ’ 3 }
  • E { 3 , 8 }

Q18:

Determine the domain and the range of 𝑓 ( π‘₯ ) = βˆ’ 1 π‘₯ + 1 .

  • A The domain is { βˆ’ 1 } and the range is ℝ .
  • B The domain is ℝ and the range is ℝ .
  • C The domain is ℝ and the range is { βˆ’ 1 } .
  • D The domain is ℝ βˆ’ { βˆ’ 1 } and the range is ℝ βˆ’ { 0 } .
  • E The domain is ℝ βˆ’ { βˆ’ 1 } and the range is ℝ βˆ’ { βˆ’ 1 } .

Q19:

Given the function 𝑛 ( π‘₯ ) = π‘₯ + 8 π‘₯ ( π‘₯ + 8 ) ( π‘₯ + 5 ) 2 2 , find the multiplicative inverse of 𝑛 in its simplest form and state its domain.

  • A 𝑛 ( π‘₯ ) = π‘₯ + 5 π‘₯ βˆ’ 1 2 , domain = ℝ βˆ’ { 0 }
  • B 𝑛 ( π‘₯ ) = π‘₯ + 5 π‘₯ βˆ’ 1 2 , domain = ℝ βˆ’ { βˆ’ 8 }
  • C 𝑛 ( π‘₯ ) = π‘₯ π‘₯ + 5 βˆ’ 1 2 , domain = ℝ βˆ’ { βˆ’ 8 }
  • D 𝑛 ( π‘₯ ) = π‘₯ + 5 π‘₯ βˆ’ 1 2 , domain = ℝ βˆ’ { βˆ’ 8 , 0 }
  • E 𝑛 ( π‘₯ ) = π‘₯ π‘₯ + 5 βˆ’ 1 2 , domain = ℝ βˆ’ { βˆ’ 8 , 0 }

Q20:

Determine the domain of the function 𝑓 ( π‘₯ ) = | π‘₯ βˆ’ 1 | | π‘₯ | βˆ’ 8 .

  • A ℝ βˆ’ { 0 }
  • B ℝ βˆ’ { 8 }
  • C ℝ βˆ’ { βˆ’ 8 }
  • D ℝ βˆ’ { βˆ’ 8 , 8 }

Q21:

Determine the domain and the range of the function 𝑓 ( π‘₯ ) = π‘₯ + 9 π‘₯ + 2 0 π‘₯ + 5 2 .

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

Q22:

Determine the domain and the range of the function 𝑓 ( π‘₯ ) = 6 π‘₯ + 1 2 π‘₯ + 2 .

  • A The domain is ℝ and the range is { 6 } .
  • B The domain is { 6 } and the range is ℝ βˆ’ { βˆ’ 2 } .
  • C The domain is { 6 } and the range is ℝ .
  • D The domain is ℝ βˆ’ { βˆ’ 2 } and the range is { 6 } .
  • E The domain is ℝ and the range is { βˆ’ 2 , 6 } .

Q23:

Determine the domain and the range of the function 𝑓 ( π‘₯ ) = βˆ’ 1 0 π‘₯ + 6 4 0 π‘₯ βˆ’ 6 4 2 2 .

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

Q24:

Determine the domain and the range of the function 𝑓 ( π‘₯ ) = 9 π‘₯ βˆ’ 8 1 π‘₯ βˆ’ 9 2 2 .

  • A The domain is ℝ and the range is { 9 } .
  • B The domain is { 9 } and the range is ℝ βˆ’ { βˆ’ 3 , 3 } .
  • C The domain is { 9 } and the range is ℝ .
  • D The domain is ℝ βˆ’ { βˆ’ 3 , 3 } and the range is { 9 } .
  • E The domain is ℝ and the range is { βˆ’ 3 , 3 } .

Q25:

Determine the domain and the range of the function 𝑓 ( π‘₯ ) = βˆ’ 5 π‘₯ βˆ’ 1 5 π‘₯ + 3 .

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

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