Lesson Worksheet: Domain and Range of a Function Mathematics

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

Q1:

Determine the range of 𝑓(𝑥).

  • A{−8,14,−2}
  • B{−20,14,15,16,18,−8,−4,−2}
  • C{−20,−4}
  • D{16,18,15}
  • E{16,18,−20,−4,15}

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[16,73)
  • B[16,73]
  • C[−4,53]
  • D[−4,53)

Q5:

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

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

Q6:

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]

Q7:

Determine the range of the function in the given figure.

  • A{5}
  • Bℝ−{−5}
  • C{−5}
  • Dℝ−{5}
  • Eℝ

Q8:

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

  • A[1,∞)
  • Bℝ−{1}
  • C[4,∞)
  • Dℝ
  • Eℝ−{4}

Q9:

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

  • A[−33,33]
  • Bℝ−[−33,33]
  • Cℝ−(−33,33)
  • D(−33,33)

Q10:

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]

Q11:

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

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

Q12:

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}

Q13:

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
  • D𝑓 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.

  • A𝑥=7 or 𝑥=3
  • B𝑥=2 or 𝑥=5
  • C𝑥=−1 or 𝑥=3
  • D𝑥=3
  • E𝑥=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?

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

Q14:

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.

Q15:

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

Q16:

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

Q17:

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

  • A{1,9,3,4}
  • B{9,21,6}
  • C{9}
  • D{11}
  • E{1,3,−3,13}

Q18:

If 𝑓∶ℝ→ℝ, where 𝑓(𝑥)=𝑥+62, find the range of 𝑓.

  • A(−∞,62)
  • B(−∞,62]
  • C(62,∞)
  • D[62,∞)

Q19:

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,−10}
  • B[−9,10]
  • C{0,1,10,−10,−9,−7,−4,−1}
  • D[−10,1]
  • E{−7,10,−4,−1,−9}

Q20:

𝑋={14,7,11}, 𝑌={33,1,42,12,21}, 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.

  • A𝑅={(14,42),(7,21)}, a function from 𝑋 to 𝑌, range ={42,21}
  • B𝑅={(14,42),(7,21),(11,33)}, a function from 𝑋 to 𝑌, range={42,21,33}
  • C𝑅={(14,42),(7,21),(11,33)}, not a function from 𝑋 to 𝑌
  • D𝑅={(14,42),(14,1),(7,21),(11,33)}, not a function from 𝑋 to 𝑌

Q21:

For two sets 𝑋 and 𝑌, a function 𝑓 exists from 𝑋 to 𝑌. Also, 𝑎∈𝑋, 𝑏∈𝑌, and 𝑎𝑅𝑏 means 𝑎 is a multiple of 𝑏. If 𝑋∪𝑌={2,3,4,15}, 𝑛(𝑋)=4, and 𝑛(𝑌)=2, find the range of the function.

  • A{2,3,4,15}
  • B{4,15}
  • C{2,4}
  • D{3,15}
  • E{2,3}

Q22:

If 𝑓 is a function from the set 𝑋 to the set 𝑌, which set contains the range of 𝑓?

  • A𝑌
  • B𝑋

Q23:

𝑓 is a function from 𝑋 to 𝑌, where 𝑎𝑅𝑏 means that 𝑏 is divisible by 𝑎 for each 𝑎∈𝑋 and 𝑏∈𝑌. If 𝑋∪𝑌={3,4,6,7,14,22,32}, 𝑛(𝑋)=3, and 𝑛(𝑋×𝑌)=12, determine the range of the function.

  • A{6}
  • B{6,14}
  • C{6,14,22,32}
  • D{32,6,14}

Q24:

𝑋={4,6,9,13}, 𝑌={𝑦∶𝑦∈ℕ,5≤𝑦<98}, ℕ is the set of natural numbers, and 𝑅 is a function from 𝑋 to 𝑌 where 𝑎𝑅𝑏 means 𝑎=13𝑏. Given that 𝑎∈𝑋 and 𝑏∈𝑌, write the range of the function.

  • A{7,9,12,16}
  • B{4,6,9,13}
  • C{12,18,27,39}
  • D{(4,12),(6,18),(9,27),(13,39)}

Q25:

If 𝑓∶{3,−6,−2,9,5}→ℝ, where 𝑓(𝑥)=−6𝑥−2, find the range of 𝑓.

  • A{−24,−16,−12,6,−2}
  • B{−16,−52,−28,38,14}
  • C{−54,−30,36,12,−18}
  • D{−56,−32,34,10,−20}
  • E{−8,1,3,−4,7}

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