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 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 .
  • B The domain is , and the range is .
  • C The domain is , and the range is { 2 } .
  • D The domain is { 2 } , and the range is { 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 = { 1 , 1 , 8 , 8 }
  • B 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 + 1 ) ( 𝑥 8 ) , domain = { 1 , 8 }
  • C 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 1 ) ( 𝑥 + 8 ) , domain = { 1 , 8 }
  • D 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 + 1 ) ( 𝑥 8 ) , domain = { 1 , 1 , 8 , 8 }
  • E 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 + 7 𝑥 8 ) ( 𝑥 1 ) ( 𝑥 6 4 ) , domain = { 1 , 1 , 8 , 8 }

Q4:

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

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

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

Q10:

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

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

Q11:

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

  • A 3 4
  • B 0 , 3 4 , 7 6
  • C 7 6
  • D 0 , 7 6
  • E { 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 { 4 , 8 }
  • C { 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 { 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 { 2 }
  • B { 1 }
  • C { 2 }
  • D { 1 }
  • E { 1 , 2 }

Q15:

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

  • A { 3 }
  • B { 9 , 3 }
  • C { 9 }
  • D { 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 .
  • B The domain is { 4 } and the range is { 2 } .
  • C The domain is and the range is { 2 } .
  • D The domain is { 2 } and the range is { 4 } .
  • E The domain is { 2 , 4 } and the range is .

Q18:

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

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

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

Q20:

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

  • A { 0 }
  • B { 8 }
  • C { 8 }
  • D { 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 .
  • B The domain is { 1 } and the range is { 5 } .
  • C The domain is and the range is { 5 } .
  • D The domain is { 5 } and the range is { 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 .
  • B The domain is { 1 0 } , and the range is { 8 , 8 } .
  • C The domain is , and the range is { 1 0 } .
  • D The domain is { 8 , 8 } , and the range is { 1 0 } .
  • E The domain is { 8 , 8 } , and the range is .

Q23:

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

  • A The domain is and the range is { 5 } .
  • B The domain is { 5 } and the range is { 3 } .
  • C The domain is { 5 } and the range is .
  • D The domain is { 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 .
  • B The domain is { 6 } and the range is { 4 } .
  • C The domain is and the range is { 4 } .
  • D The domain is { 4 } and the range is { 6 } .
  • E The domain is { 6 , 4 } and the range is .

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