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 their equations.

Q1:

Determine the range of 𝑓(𝑥).

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

Q2:

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

  • A [ − 2 , 2 ]
  • B { 0 , − 4 , − 3 , − 2 , − 1 }
  • C { 0 , 1 , 2 , − 2 , − 1 }
  • D { 0 , 1 , 2 , − 4 , − 3 , − 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 { 4 , 9 }
  • B { 4 , 4 , 9 }
  • C { − 9 , − 6 , − 5 }
  • D { 3 , 4 , 6 , 9 }
  • E { ( − 9 , 4 ) , ( − 6 , 4 ) , ( − 5 , 9 ) }

Q4:

If 𝑓∶[2,21]⟶ℝ, where 𝑓(𝑥)=3𝑥−10, 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√2𝑥−1.

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

Q7:

The figure below shows the graph of a function 𝑓.

What is the range of the function?

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

Q8:

Determine the domain of the function 𝑓(𝑥)=𝑥+4𝑥∈[−4,4],−8𝑥+40𝑥∈(4,5].ifif

  • 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−𝑥4≤𝑥≤8.ifif

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

Q10:

Determine the range of the function in the given figure.

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

Q11:

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

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

Q12:

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

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

Q13:

Determine the domain of the function 𝑓(𝑥)=√|𝑥|−33.

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

Q14:

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

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

Q15:

Determine the domain of the function 𝑓(𝑥)=1|𝑥|−65.

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

Q17:

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

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

  • A3
  • B7
  • C1
  • D 𝑓 is not defined at 0.
  • E2

What is the domain of 𝑓?

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

What is the range of 𝑓?

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

Solve 𝑓(𝑥)=3.

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

What is the solution set of 𝑓(𝑥)=−1?

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

What is the solution set of 𝑓(𝑥)=0?

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

Q19:

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

  • AThe domain is ℝ, and the range is ℝ.
  • BThe domain is ℝ−{−9}, and the range is ℝ−{8}.
  • CThe domain is ℝ, and the range is (−∞,8].
  • DThe domain is ℝ−{8}, and the range is ℝ−{−9}.
  • EThe domain is (−∞,8], and the range is ℝ.

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.
  • Aa only
  • Ba and c
  • Cb only
  • Dc only

Q21:

What is the range of the function 𝑓(𝑥)=𝑥+12 on the domain ℤ (which is the domain of all of the integers)?

  • Aall the integers
  • Ball the numbers of the form𝑛2 where 𝑛 is an integer
  • CWe cannot tell the range because we do not know the codomain.
  • Dall the positive integers
  • Eall the rational numbers

Q22:

If 𝑓∶{−4,−1,4,−2}→[6,25] and 𝑓(𝑥)=𝑥+5, find the range of 𝑓.

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

Q23:

If 𝑓∶ℝ→ℝ, where 𝑓(𝑥)=𝑥+62, 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≤10} and 𝑓𝑋→ℝ:, where the graph of the function can be represented by 𝐺={(10,1),(−7,−1),(−4,−10),(−1,−10),(−9,0)}, determine its range.

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

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 𝑎𝑅𝑏 means that 𝑎=13𝑏 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 ={42,21}
  • B 𝑅 = { ( 1 4 , 4 2 ) , ( 7 , 2 1 ) , ( 1 1 , 3 3 ) } , a function from 𝑋 to 𝑌, range={42,21,33}
  • 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 ) , ( 1 4 , 1 ) , ( 7 , 2 1 ) , ( 1 1 , 3 3 ) } , not a function from 𝑋 to 𝑌

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