Worksheet: Recognizing Functions from Graphs

In this worksheet, we will practice recognizing functions from schematic descriptions, arrow diagrams, and graphs.

Q1:

Does the given graph represent a function?

  • Ayes
  • Bno

Q2:

Fady believes that setting 𝑓 ( 𝑥 ) equal to the digit before the decimal point in the decimal expansion of 𝑥 , for each real number, defines a function from the real numbers to the set of digits { 0 , 1 , , 9 } . Since 1 . ̇ 0 and 0 . ̇ 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 .

Q3:

Let 𝑋 = and 𝑌 = . Which of the following properties is true of the relation between 𝑋 and 𝑌 given by 𝑦 = 𝑥 3 2 , where 𝑥 𝑋 and 𝑦 𝑌 ?

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

Q4:

If 𝑓 is a function from the set 𝑋 to the set 𝑌 , what do we call 𝑌 ?

  • 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

Q5:

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

  • A(C)
  • B(B)
  • C(D)
  • D(A)

Q6:

If 𝑋 = { 5 , 3 , 6 } , 𝑛 ( 𝑌 ) = 4 , and the function 𝑓 𝑋 𝑌 , where 𝑓 ( 𝑥 ) = 𝑥 2 𝑥 + 5 , which of the sets below can be a representation of 𝑌 ?

  • 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 }

Q7:

For two sets 𝑋 and 𝑌 , a function 𝑓 exists from 𝑋 to 𝑌 . Also, 𝑎 𝑋 , 𝑏 𝑌 , and 𝑎 𝑅 𝑏 means 𝑎 is a multiple of 𝑏 . If 𝑋 𝑌 = { 2 , 6 , 7 , 3 5 } , 𝑛 ( 𝑋 ) = 4 , and 𝑛 ( 𝑌 ) = 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 ) }

Q8:

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

  • A 𝑋 = { 8 , 1 0 } , 𝑌 = { 4 , 5 }
  • B 𝑋 = { 4 , 5 } , 𝑌 = { 8 , 1 0 }
  • C 𝑋 = { 8 , 1 0 } , 𝑌 = { 4 , 5 , 8 , 1 0 }
  • D 𝑋 = { 4 , 5 , 8 , 1 0 } , 𝑌 = { 4 , 5 }
  • E 𝑋 = { 4 , 5 , 8 , 1 0 } , 𝑌 = { 8 , 1 0 }

Q9:

For two sets 𝑋 and 𝑌 , a function 𝑓 exists from 𝑋 to 𝑌 . Also, 𝑎 𝑋 , 𝑏 𝑌 , and 𝑎 𝑅 𝑏 means 𝑏 is divisible by 𝑎 . If 𝑋 𝑌 = { 4 , 6 , 7 , 8 , 1 8 , 2 1 , 2 9 } , 𝑛 ( 𝑋 ) = 3 , and 𝑛 ( 𝑋 × 𝑌 ) = 1 2 , find 𝑋 and 𝑌 .

  • A 𝑋 = { 8 , 1 8 , 2 1 } , 𝑌 = { 4 , 6 , 7 }
  • B 𝑋 = { 4 , 6 , 7 } , 𝑌 = { 8 , 1 8 , 2 1 }
  • C 𝑋 = { 8 , 1 8 , 2 1 , 2 9 } , 𝑌 = { 4 , 6 , 7 }
  • D 𝑋 = { 4 , 6 , 7 } , 𝑌 = { 8 , 1 8 , 2 1 , 2 9 }

Q10:

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

  • Atrue
  • Bfalse

Q11:

Can the equation 𝑥 + 𝑦 = 4 2 2 be expressed as a function? If yes, state the function.

  • AYes, 𝑓 ( 𝑥 ) = ± 4 𝑥 2
  • BYes, 𝑓 ( 𝑥 ) = 4 𝑥 2
  • CYes, 𝑓 ( 𝑥 ) = ± 2 𝑥 2
  • DNo
  • EYes, 𝑓 ( 𝑥 ) = 2 𝑥 2

Q12:

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

Q13:

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

  • AIt is a function, 𝑥 = 4 𝑦
  • B It is a function, 𝑦 = 4 𝑥
  • CIt is a function, 𝑥 = 4 𝑥
  • DIt is not a function.
  • EIt is a function, 𝑦 = 4 𝑦

Q14:

Which of the following equations are NOT functions of 𝑥 ?

  1. 𝑦 = 𝑥 3
  2. 𝑥 = 7
  3. 𝑦 = 𝑥 + 4 , 𝑥 0
  4. 𝑓 ( 𝑦 ) = 2 𝑦 3
  • Ab and c
  • Ba and c
  • Ca and d
  • Db and d
  • Ea and b

Q15:

Which of the following is the equation 𝑦 = 4 𝑥 + 7 expressed in function notation?

  • AThis cannot be expressed as a function.
  • B 4 𝑥 + 7 = 0
  • C 𝑥 = 4 𝑦 + 7
  • D 𝑓 ( 𝑥 ) = 4 𝑥 + 7
  • E 𝑦 = 7 𝑥 + 4

Q16:

Determine whether the following statement is true or false: The given figure represents a function if 𝑥 is the input and 𝑦 is the output.

  • Afalse
  • Btrue

Q17:

If 𝑓 is a function from the set 𝑋 to the set 𝑌 , what do we call 𝑋 ?

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

Q18:

Given that 𝑋 = { 7 , 1 , 9 } , and 𝑅 = { ( 𝑎 , 1 ) , ( 𝑏 , 7 ) , ( 7 , 9 ) } , where 𝑅 is a function on 𝑋 , find the numerical value of 𝑎 + 𝑏 .

Q19:

What is the codomain of a function?

  • Aa set of values that can go into the function
  • Ba set of values that actually comes out of the function
  • Ca set of values that may possibly come out of or go into the function
  • Da set of values in which the outputs of a function are contained
  • Ea set of values that actually comes out of or goes into the function

Q20:

The figure below shows the graph of a function 𝑓 .

Which of the following cannot be the codomain of this function?

  • A the set of rational numbers
  • B the set of real numbers
  • C the interval [ 3 , 1 ]
  • D the set of negative integers
  • Ethe numbers 𝑦 such that 3 𝑦 0

Q21:

Given that 𝑓 𝑋 𝑌 , 𝑋 = { 5 , 6 , 8 } , 𝑌 = { 5 , 3 , 0 , 2 } , and 𝑓 = { ( 5 , 2 ) , ( 6 , 3 ) , ( 8 , 5 ) } , find the codomain of the function.

  • A { ( 5 , 2 ) , ( 6 , 3 ) , ( 8 , 5 ) }
  • B { 5 , 6 , 8 }
  • C { 5 , 3 , 2 }
  • D { 5 , 3 , 0 , 2 }

Q22:

𝑋 = { 3 , 4 , 7 , 2 } , 𝑓 ( 3 ) = 4 , 𝑓 ( 4 ) = 8 , 𝑓 ( 7 ) = 5 , and 𝑓 ( 2 ) = 2 where 𝑓 is a function on 𝑋 . Find the ordered pairs that satisfy the function and its range.

  • A 𝑓 = { ( 3 , 8 ) , ( 4 , 4 ) , ( 7 , 2 ) , ( 2 , 5 ) } , range = { 8 , 4 , 2 , 5 }
  • B 𝑓 = { ( 4 , 3 ) , ( 8 , 4 ) , ( 5 , 7 ) , ( 2 , 2 ) } , range = { 3 , 4 , 7 , 2 }
  • C 𝑓 = { ( 4 , 4 ) , ( 8 , 3 ) , ( 5 , 2 ) , ( 2 , 7 ) } , range = { 4 , 3 , 2 , 7 }
  • D 𝑓 = { ( 3 , 4 ) , ( 4 , 8 ) , ( 7 , 5 ) , ( 2 , 2 ) } , range = { 4 , 8 , 5 , 2 }
  • E 𝑓 = { ( 4 , 4 ) , ( 8 , 3 ) , ( 2 , 2 ) , ( 5 , 7 ) } , range = { 3 , 4 , 2 , 7 }

Q23:

Suppose 𝐴 = { 𝑎 } and 𝐵 is a set with 5 elements. How many different functions are there from 𝐴 to 𝐵 ?

Q24:

Suppose 𝑋 is a set with 𝑛 elements and 𝑌 is a set with 𝑚 elements, where 𝑛 0 , 𝑚 0 , and 𝑛 𝑚 . How many different functions are there from 𝑋 to 𝑌 ?

  • A 𝑛 𝑚
  • B 𝑛 𝑚
  • C 𝑛 + 𝑚
  • D 𝑚 𝑛
  • EIt is not possible to say.

Q25:

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

  • A 𝑋 × 𝑌
  • B 𝑌
  • C 𝑌 × 𝑋
  • D 𝑋

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