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Lesson: Domain and Range of a Rational Function

In this lesson, we will learn how to find the domain and range of a rational function either from its graph or its defining rule.

Sample Question Videos

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Worksheet • 25 Questions • 3 Videos

Q1:

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’ 6 5
  • B14
  • C 6 5
  • D 5 6
  • E βˆ’ 5 6

Q2:

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

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

Q3:

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

Q4:

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

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

Q5:

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

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

Q6:

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

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

Q7:

Define a function on real numbers by 𝑓 ( π‘₯ ) = 2 π‘₯ + 3 4 π‘₯ + 5 .

What is the domain of this function?

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

Find the one value that 𝑓 ( π‘₯ ) cannot take.

  • A 1 2
  • B βˆ’ 1 2
  • C βˆ’ 5 4
  • D βˆ’ 3 2
  • E 5 4

What is the range of this function?

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

Q8:

Given that the domain of the function 𝑛 ( π‘₯ ) = 3 6 π‘₯ + 2 0 π‘₯ + π‘Ž is ℝ βˆ’ { βˆ’ 2 , 0 } , evaluate 𝑛 ( 3 ) .

Q9:

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

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

Q10:

Identify the domain of .

  • A
  • B
  • C
  • D
  • E

Q11:

Determine the domain of the function 𝑛 ( π‘₯ ) = 4 π‘₯ + 3 π‘₯ 6 π‘₯ + 7 π‘₯ 2 2 .

  • A ℝ βˆ’  0 , βˆ’ 7 6 
  • B ℝ βˆ’ { 0 }
  • C ℝ βˆ’  0 , βˆ’ 3 4 , βˆ’ 7 6 
  • D ℝ βˆ’  βˆ’ 7 6 
  • E ℝ βˆ’  βˆ’ 3 4 

Q12:

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

Q13:

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 ?

Q14:

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

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

Q15:

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

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

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 ( π‘₯ ) = π‘₯ βˆ’ 4 π‘₯ βˆ’ 2 2 .

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

Q18:

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

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

Q20:

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

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

Q21:

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

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

Q22:

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

Q23:

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

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

Q24:

Given that the domain of the function 𝑛 ( π‘₯ ) = 8 π‘₯ βˆ’ 9 1 is the same as the domain of the function 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 5 π‘₯ + 𝑙 2 , what is the value of 𝑙 ?

Q25:

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

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