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Worksheet: Definition of Functions

Q1:

Does the given graph represent a function?

  • Ayes
  • Bno

Q2:

Does the given graph represent a function?

  • Ayes
  • Bno

Q3:

Does the given graph represent a function?

  • Ano
  • Byes

Q4:

Does the given graph represent a function?

  • Ano
  • Byes

Q5:

Does the given graph represent a function?

  • Ano
  • Byes

Q6:

Does the given graph represent a function?

  • Ano
  • Byes

Q7:

Does the given graph represent a function?

  • Ayes
  • Bno

Q8:

Does the given graph represent a function?

  • Ano
  • Byes

Q9:

Does the given graph represent a function?

  • Ayes
  • Bno

Q10:

Does the given graph represent a function?

  • Ayes
  • Bno

Q11:

Does the given graph represent a function?

  • Ayes
  • Bno

Q12:

Does the given graph represent a function?

  • Ayes
  • Bno

Q13:

Michael believes that setting 𝑓 ( π‘₯ ) equal to the digit before the decimal point in the decimal expansion of π‘₯ , for each real number, defines a function from the real numbers ℝ to the set of digits { 0 , 1 , … , 9 } . Since 1 . 0 0 0 and 0 . 9 9 9 are both decimal expansions of the real number 1, what does that say about 𝑓 ?

  • A 𝑓 defines a function with 𝑓 ( 1 ) = 1 .
  • B 𝑓 defines a function with 𝑓 ( 1 ) = 0 .
  • C 𝑓 defines a function with 𝑓 ( 1 ) = 0 or 1.
  • D 𝑓 does not define a function.
  • E 𝑓 defines a function with 𝑓 ( 0 ) = 1 .

Q14:

Let 𝑋 = ℝ and π‘Œ = ℝ . Which of the following properties is true of the relation between 𝑋 and π‘Œ given by 𝑦 = π‘₯ 3 2 , where π‘₯ ∈ 𝑋 and 𝑦 ∈ π‘Œ ?

  • A 𝑦 is not a function of π‘₯ because when 𝑦 = βˆ’ 1 there is no corresponding value of π‘₯ .
  • B 𝑦 is not a function of π‘₯ because 3 √ π‘₯ 2 cannot always be found.
  • C 𝑦 is not a function of π‘₯ because when 𝑦 = 1 there is no corresponding value of π‘₯ .
  • D 𝑦 is a function of π‘₯ because each input π‘₯ gives a real number whose cube root gives the unique output 𝑦 .

Q15:

If 𝑓 is a function from the set 𝑋 to the set π‘Œ , what do we call π‘Œ ?

  • A the range of the function 𝑓
  • B the domain of the function 𝑓
  • C the rule of the function 𝑓
  • D the codomain of the function 𝑓

Q16:

If 𝑋 = { βˆ’ 6 , βˆ’ 9 , 0 , 2 } , which of the following arrow diagrams represents a function on the set 𝑋 ?

  • A(C)
  • B(B)
  • C(D)
  • D(A)

Q17:

If 𝑋 = { 5 , 3 , 6 } , 𝑛 ( π‘Œ ) = 4 , and the function 𝑓 ∢ 𝑋 β†’ π‘Œ , where 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 2 π‘₯ + 5 2 , which of the sets below can be a representation of π‘Œ ?

  • A { 2 0 , 8 , 4 0 , 5 }
  • B { 2 0 , 8 , 4 0 }
  • C { 2 0 , 8 , 2 9 }
  • D { 2 0 , 8 , 2 9 , 4 0 }

Q18:

For two sets 𝑋 and π‘Œ , a function 𝑓 exists from 𝑋 to π‘Œ . Also, π‘Ž ∈ 𝑋 , 𝑏 ∈ π‘Œ , and π‘Ž 𝑅 𝑏 means π‘Ž is a multiple of 𝑏 . If 𝑋 βˆͺ π‘Œ = { 2 , 6 , 7 , 3 5 } , 𝑛 ( 𝑋 ) = 4 , and 𝑛 ( π‘Œ ) = 2 , determine 𝑅 .

  • A 𝑅 = { ( 6 , 2 ) , ( 3 5 , 7 ) }
  • B 𝑅 = { ( 2 , 2 ) , ( 2 , 6 ) , ( 7 , 7 ) , ( 7 , 3 5 ) }
  • C 𝑅 = { ( 2 , 6 ) , ( 7 , 3 5 ) }
  • D 𝑅 = { ( 2 , 2 ) , ( 6 , 2 ) , ( 7 , 7 ) , ( 3 5 , 7 ) }
  • E 𝑅 = { ( 2 , 2 ) , ( 6 , 2 ) , ( 6 , 6 ) , ( 7 , 7 ) , ( 3 5 , 7 ) , ( 3 5 , 3 5 ) }

Q19:

For two sets 𝑋 and π‘Œ , a function 𝑓 exists from 𝑋 to π‘Œ . Also, π‘Ž ∈ 𝑋 , 𝑏 ∈ π‘Œ , and π‘Ž 𝑅 𝑏 means π‘Ž is a multiple of 𝑏 . If 𝑋 βˆͺ π‘Œ = { 4 , 5 , 8 , 1 0 } , 𝑛 ( 𝑋 ) = 4 , and 𝑛 ( π‘Œ ) = 2 , which of the following definitions of 𝑋 and π‘Œ are correct?

  • A 𝑋 = { 8 , 1 0 } , π‘Œ = { 4 , 5 }
  • B 𝑋 = { 4 , 5 } , π‘Œ = { 8 , 1 0 }
  • C 𝑋 = { 8 , 1 0 } , π‘Œ = { 4 , 5 , 8 , 1 0 }
  • D 𝑋 = { 4 , 5 , 8 , 1 0 } , π‘Œ = { 4 , 5 }
  • E 𝑋 = { 4 , 5 , 8 , 1 0 } , π‘Œ = { 8 , 1 0 }

Q20:

For two sets 𝑋 and π‘Œ , a function 𝑓 exists from 𝑋 to π‘Œ . Also, π‘Ž ∈ 𝑋 , 𝑏 ∈ π‘Œ , and π‘Ž 𝑅 𝑏 means 𝑏 is divisible by π‘Ž . If 𝑋 βˆͺ π‘Œ = { 4 , 6 , 7 , 8 , 1 8 , 2 1 , 2 9 } , 𝑛 ( 𝑋 ) = 3 , and 𝑛 ( 𝑋 Γ— π‘Œ ) = 1 2 , find 𝑋 and π‘Œ .

  • A 𝑋 = { 8 , 1 8 , 2 1 } , π‘Œ = { 4 , 6 , 7 }
  • B 𝑋 = { 4 , 6 , 7 } , π‘Œ = { 8 , 1 8 , 2 1 }
  • C 𝑋 = { 8 , 1 8 , 2 1 , 2 9 } , π‘Œ = { 4 , 6 , 7 }
  • D 𝑋 = { 4 , 6 , 7 } , π‘Œ = { 8 , 1 8 , 2 1 , 2 9 }

Q21:

Determine whether the following statement is true or false: The shown figure represents a function.

  • Atrue
  • Bfalse

Q22:

Can the equation π‘₯ + 𝑦 = 4 2 2 be expressed as a function? If yes, state the function.

  • AYes, 𝑓 ( π‘₯ ) = Β± √ 4 βˆ’ π‘₯ 2
  • BYes, 𝑓 ( π‘₯ ) = √ 4 βˆ’ π‘₯ 2
  • CYes, 𝑓 ( π‘₯ ) = Β± √ 2 βˆ’ π‘₯ 2
  • DNo
  • EYes, 𝑓 ( π‘₯ ) = √ 2 βˆ’ π‘₯ 2

Q23:

What is a function?

  • AAn independent quantity
  • BA dependent quantity
  • CA combination of domain and range
  • DA relation that relates each input to exactly one output

Q24:

Given that π‘₯ and 𝑦 are variables, determine whether 𝑓 ( π‘₯ ) = 4 ( 𝑦 ) is a function, and if it is, state which equation is equivalent to it.

  • AIt is a function, π‘₯ = 4 𝑦
  • B It is a function, 𝑦 = 4 π‘₯
  • CIt is a function, π‘₯ = 4 π‘₯
  • DIt is not a function.
  • EIt is a function, 𝑦 = 4 𝑦

Q25:

Which of the following equations are NOT functions of π‘₯ ?

  1. 𝑦 = π‘₯ 3
  2. π‘₯ = 7
  3. 𝑦 = √ π‘₯ + 4 , π‘₯ β‰₯ 0
  4. 𝑓 ( 𝑦 ) = 2 𝑦 βˆ’ 3
  • Ab and c
  • Ba and c
  • Ca and d
  • Db and d
  • Ea and b