Worksheet: Domain and Range of a Function

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In this worksheet, we will practice identifying the domain and range of functions from schematics, graphs, and equations.

Q1:

Determine the range of 𝑓 ( 𝑥 ) .

  • A { − 8 , 1 4 , − 2 }
  • B { 1 6 , 1 8 , − 2 0 , − 4 , 1 5 }
  • C { − 2 0 , 1 4 , 1 5 , 1 6 , 1 8 , − 8 , − 4 , − 2 }
  • D { − 2 0 , − 4 }
  • E { 1 6 , 1 8 , 1 5 }

Q2:

Using the figure below, determine the range of 𝑓 ( 𝑥 ) .

  • A { 0 , 1 , 2 , − 4 , − 3 , − 2 , − 1 }
  • B { 0 , − 4 , − 3 , − 2 , − 1 }
  • C [ − 2 , 2 ]
  • D { 0 , 1 , 2 , − 2 , − 1 }
  • E [ − 4 , 0 ]

Q3:

Given that 𝑓 ∶ 𝑋 → 𝑌 , 𝑋 = { − 9 , − 6 , − 5 } , 𝑌 = { 3 , 4 , 6 , 9 } , and 𝑓 = { ( − 9 , 4 ) , ( − 6 , 4 ) , ( − 5 , 9 ) } , which of the following sets correctly specify the range of the function?

  • A { ( − 9 , 4 ) , ( − 6 , 4 ) , ( − 5 , 9 ) }
  • B { 3 , 4 , 6 , 9 }
  • C { 4 , 4 , 9 }
  • D { 4 , 9 }
  • E { − 9 , − 6 , − 5 }

Q4:

If 𝑓 ∶ [ 2 , 2 1 ] ⟶ ℝ , where 𝑓 ( 𝑥 ) = 3 𝑥 − 1 0 , find the range of 𝑓 .

  • A [ 1 6 , 7 3 ]
  • B [ 1 6 , 7 3 )
  • C [ − 4 , 5 3 )
  • D [ − 4 , 5 3 ]

Q5:

Find the domain of the function 𝑓 ( 𝑥 ) = √ 𝑥 .

  • A ( 0 , ∞ )
  • B ℝ
  • C ] − ∞ , 0 ]
  • D [ 0 , ∞ )

Q6:

Find the domain of the function 𝑓 ( 𝑥 ) = − 3 √ 2 𝑥 − 1 .

  • A ℝ − 1 2 , ∞
  • B − ∞ , 1 2
  • C − 1 2 , ∞
  • D 1 2 , ∞
  • E ℝ − 1 2

Q7:

The figure below shows the graph of a function 𝑓 .

What is the range of the function?

  • A { − 3 , − 2 , − 1 , 0 , 1 }
  • B { − 3 , − 2 , − 1 , 0 }
  • C the interval [ − 3 , 0 ]
  • D { − 3 , − 2 , 0 }
  • EIt is not possible to determine the range.

Q8:

Determine the domain of the function 𝑓 ( 𝑥 ) = 𝑥 + 4 𝑥 ∈ [ − 4 , 4 ] , − 8 𝑥 + 4 0 𝑥 ∈ ] 4 , 5 ] . i f i f

  • A { 0 , 8 }
  • B [ 0 , 8 ]
  • C ] − 4 , 5 [
  • D [ − 4 , 5 ]

Q9:

Determine the range of the function 𝑓 ( 𝑥 ) = 𝑥 − 2 ≤ 𝑥 < 4 , 8 − 𝑥 4 ≤ 𝑥 ≤ 8 . i f i f

  • A ℝ − { − 2 , 4 }
  • B ℝ − { 0 , 4 }
  • C ( − 2 , 4 ]
  • D [ − 2 , 4 ]
  • E { − 2 , 8 }

Q10:

Determine the range of the function in the given figure.

  • A ℝ − { − 5 }
  • B { 5 }
  • C ℝ
  • D { − 5 }
  • E ℝ − { 5 }

Q11:

Determine the domain of the function represented in the graph below.

  • A ℝ − { 4 }
  • B [ 1 , ∞ )
  • C ℝ − { 1 }
  • D [ 4 , ∞ )
  • E ℝ

Q12:

Find the range of the function 𝑓 ( 𝑥 ) = 𝑥 𝑥 ∈ [ 0 , 3 ) , 3 𝑥 ∈ [ 3 , 5 ] , 8 − 𝑥 𝑥 ∈ ( 5 , 8 ] . i f i f i f

  • A ℝ − { 0 , 3 }
  • B [ 0 , 8 ]
  • C ℝ − { 0 , 8 }
  • D [ 0 , 3 ]
  • E { 0 , 8 }

Q13:

Determine the domain of the function 𝑓 ( 𝑥 ) = √ | 𝑥 | − 3 3 .

  • A ℝ − [ − 3 3 , 3 3 ]
  • B ] − 3 3 , 3 3 [
  • C [ − 3 3 , 3 3 ]
  • D ℝ − ] − 3 3 , 3 3 [

Q14:

Determine the domain of the function 𝑓 ( 𝑥 ) = √ 𝑥 + 3 − √ 8 − 𝑥 .

  • A [ − 3 , 8 )
  • B ( − 3 , 8 )
  • C [ 8 , ∞ )
  • D [ − 3 , 8 ]
  • E ( − ∞ , − 3 ]

Q15:

Determine the domain of the function 𝑓 ( 𝑥 ) = 1 | 𝑥 | − 6 5 .

  • A [ − 6 5 , 6 5 ]
  • B { − 6 5 , 6 5 }
  • C ℝ − [ − 6 5 , 6 5 ]
  • D ℝ − { − 6 5 , 6 5 }

Q16:

Determine the range of the function represented in the figure below.

  • A { 𝑎 , 𝑐 }
  • B { 𝑎 , 𝑏 }
  • C { 𝑏 , 𝑐 }
  • D { 𝑎 , 𝑏 , 𝑐 }

Q17:

Determine the range of the function represented by the figure in 𝑋 .

  • A { − 5 , − 4 }
  • B { − 5 , − 4 , − 3 , − 2 }
  • C { − 5 , − 2 }
  • D { − 5 , − 4 , − 3 }
  • E { − 3 }

Q18:

Given the function 𝑓 = { ( 2 , 1 ) , ( 1 , 3 ) , ( 3 , − 1 ) , ( 7 , 3 ) } , answer the following.

Evaluate 𝑓 ( 2 ) .

Evaluate 𝑓 ( 3 ) .

Evaluate 𝑓 ( 0 ) .

  • A2
  • B1
  • C 𝑓 is not defined at 0.
  • D3
  • E7

What is the domain of 𝑓 ?

  • A { 1 , 2 , 3 , 7 }
  • B { − 1 , 1 , 3 }
  • C { 1 , 3 , 7 }
  • D { 1 , 2 , 3 }
  • E { 0 , 1 , 2 , 3 }

What is the range of 𝑓 ?

  • A { 1 , 2 , 3 , 7 }
  • B { − 1 , 1 , 2 }
  • C { 0 , − 1 , 1 , 3 }
  • D { − 1 , 1 , 3 }
  • E { 0 , 1 , 2 , 3 }

Solve 𝑓 ( 𝑥 ) = 3 .

  • A 𝑥 = 3
  • B 𝑥 = − 1 or 𝑥 = 3
  • C 𝑥 = 2 or 𝑥 = 5
  • D 𝑥 = 1 or 𝑥 = 7
  • E 𝑥 = 7 or 𝑥 = 3

What is the solution set of 𝑓 ( 𝑥 ) = − 1 ?

  • A { 7 }
  • B { 3 }
  • C { − 1 }
  • D { 2 }
  • E { 1 }

What is the solution set of 𝑓 ( 𝑥 ) = 0 ?

  • A { 0 }
  • B { 7 }
  • C { 2 }
  • D { }
  • E { − 1 }

Q19:

Determine the domain and the range of the function 𝑓 ( 𝑥 ) = − 6 ( 𝑥 + 9 ) + 8 .

  • A The domain is ℝ − { − 9 } , and the range is ℝ − { 8 } .
  • B The domain is ( − ∞ , 8 ] , and the range is ℝ .
  • C The domain is ℝ − { 8 } , and the range is ℝ − { − 9 } .
  • D The domain is ℝ , and the range is ℝ .
  • E The domain is ℝ , and the range is ( − ∞ , 8 ] .

Q20:

Functions 𝑓 and 𝑔 have the same domain {1, 2, 3, 4}. Which of the following statements must be true?

  1. The range of 𝑓 + 𝑔 has at most 4 elements.
  2. The range of 𝑓 + 𝑔 has 𝑛 + 𝑚 elements, where 𝑛 = the size of the range of 𝑓 and 𝑚 = the size of the range of 𝑔 .
  3. The range of 𝑓 + 𝑔 has at least 1 element.
  • A c only
  • B a only
  • C b only
  • D a and c

Q21:

What is the range of the function 𝑓 ( 𝑥 ) = 𝑥 + 1 2 on the domain ℤ (which is the domain of all of the integers)?

  • Aall the integers
  • BWe cannot tell the range because we do not know the codomain.
  • Call the positive integers
  • Dall the numbers of the form 𝑛 2 where 𝑛 is an integer
  • Eall the rational numbers

Q22:

If 𝑓 ∶ { − 4 , − 1 , 4 , − 2 } → [ 6 , 2 5 ] and 𝑓 ( 𝑥 ) = 𝑥 + 5 , find the range of 𝑓 .

  • A { 9 }
  • B { 1 , 3 , − 3 , 1 3 }
  • C { 1 , 9 , 3 , 4 }
  • D { 9 , 2 1 , 6 }
  • E { 1 1 }

Q23:

If 𝑓 ∶ ℝ → ℝ , where 𝑓 ( 𝑥 ) = 𝑥 + 6 2 , find the range of 𝑓 .

  • A ( − ∞ , 6 2 )
  • B [ 6 2 , ∞ )
  • C ( − ∞ , 6 2 ]
  • D ( 6 2 , ∞ )

Q24:

Given that 𝑋 = { 𝑥 ∶ 𝑥 ∈ ℤ , − 9 ≤ 𝑥 ≤ 1 0 } and 𝑓 𝑋 → ℝ : , where the graph of the function can be represented by 𝐺 = { ( 1 0 , 1 ) , ( − 7 , − 1 ) , ( − 4 , − 1 0 ) , ( − 1 , − 1 0 ) , ( − 9 , 0 ) } , determine its range.

  • A { 0 , 1 , 1 0 , − 1 0 , − 9 , − 7 , − 4 , − 1 }
  • B { − 7 , 1 0 , − 4 , − 1 , − 9 }
  • C [ − 1 0 , 1 ]
  • D { 0 , 1 , − 1 , − 1 0 }
  • E [ − 9 , 1 0 ]

Q25:

𝑋 = { 1 4 , 7 , 1 1 } , 𝑌 = { 3 3 , 1 , 4 2 , 1 2 , 2 1 } , and 𝑅 is a relation from 𝑋 to 𝑌 , where 𝑎 𝑅 𝑏 means that 𝑎 = 1 3 𝑏 for each 𝑎 ∈ 𝑋 and 𝑏 ∈ 𝑌 . Determine the relation 𝑅 , state whether it represents a function or not, and write its range if it is a function.

  • A 𝑅 = { ( 1 4 , 4 2 ) , ( 7 , 2 1 ) } , a function from 𝑋 to 𝑌 , range = { 4 2 , 2 1 }
  • B 𝑅 = { ( 1 4 , 4 2 ) , ( 1 4 , 1 ) , ( 7 , 2 1 ) , ( 1 1 , 3 3 ) } , not a function from 𝑋 to 𝑌
  • C 𝑅 = { ( 1 4 , 4 2 ) , ( 7 , 2 1 ) , ( 1 1 , 3 3 ) } , not a function from 𝑋 to 𝑌
  • D 𝑅 = { ( 1 4 , 4 2 ) , ( 7 , 2 1 ) , ( 1 1 , 3 3 ) } , a function from 𝑋 to 𝑌 , range = { 4 2 , 2 1 , 3 3 }

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