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Lesson: Inverse and Composite Functions

Sample Question Videos

Worksheet • 8 Questions • 2 Videos

Q1:

Which of the following functions is NOT its own inverse?

  • A 𝑓 ( π‘₯ ) = 8 π‘₯
  • B 𝑓 ( π‘₯ ) = βˆ’ 8 βˆ’ π‘₯
  • C 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯
  • D 𝑓 ( π‘₯ ) = βˆ’ 8 π‘₯
  • E 𝑓 ( π‘₯ ) = π‘₯

Q2:

Suppose 𝑔 ( π‘₯ ) = π‘Ž π‘₯ + 𝑏 is the inverse function to 𝑓 ( π‘₯ ) = 2 βˆ’ 3 π‘₯ . Determine π‘Ž and 𝑏 by considering the composition 𝑓 ( 𝑔 ( π‘₯ ) ) .

  • A π‘Ž = βˆ’ 1 3 , 𝑏 = 2 3
  • B π‘Ž = 1 3 , 𝑏 = βˆ’ 2 3
  • C π‘Ž = 1 3 , 𝑏 = 2 3
  • D π‘Ž = 2 3 , 𝑏 = βˆ’ 1 3
  • E π‘Ž = βˆ’ 2 3 , 𝑏 = 1 3

Q3:

Let 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 1 3 π‘₯ βˆ’ 2 . Determine which of the following functions 𝑔 is the inverse of 𝑓 by checking whether 𝑓 ( 𝑔 ( π‘₯ ) ) = π‘₯ .

  • A 𝑔 ( π‘₯ ) = 2 π‘₯ βˆ’ 1 6 π‘₯ βˆ’ 4
  • B 𝑔 ( π‘₯ ) = 2 π‘₯ βˆ’ 1 3 π‘₯ βˆ’ 2
  • C 𝑔 ( π‘₯ ) = βˆ’ π‘₯ βˆ’ 1 3 π‘₯ βˆ’ 2

Q4:

Determine π‘Ž and 𝑏 so that 𝑔 ( π‘₯ ) = π‘Ž + 𝑏 √ π‘₯ is an inverse function to 𝑓 ( π‘₯ ) = ( π‘₯ βˆ’ 3 )  by considering 𝑓 ( 𝑔 ( π‘₯ ) ) .

  • A π‘Ž = 3 , 𝑏 = 1 and π‘Ž = 3 , 𝑏 = βˆ’ 1
  • B π‘Ž = βˆ’ 3 , 𝑏 = βˆ’ 1 and π‘Ž = 3 , 𝑏 = βˆ’ 1
  • C π‘Ž = 3 , 𝑏 = 1
  • D π‘Ž = βˆ’ 3 , 𝑏 = 1 and π‘Ž = 3 , 𝑏 = βˆ’ 1
  • E π‘Ž = 3 , 𝑏 = βˆ’ 1

Q5:

Shown are the graphs of functions 𝑓 and g from the set { 1 , 2 , 3 , 4 , 5 , 6 } into itself. Write the function β„Ž ( π‘₯ ) = 𝑓 ( 𝑔 ( π‘₯ ) ) as a list of ordered pairs. What can you conclude?

  • A β„Ž = { ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) , ( 4 , 4 ) , ( 5 , 5 ) , ( 6 , 6 ) } , 𝑔 is the inverse of 𝑓
  • B β„Ž = { ( 1 , 2 ) , ( 2 , 2 ) , ( 3 , 5 ) , ( 4 , 4 ) , ( 5 , 1 ) , ( 6 , 6 ) } , 𝑓 is the inverse of β„Ž
  • C β„Ž = { ( 1 , 2 ) , ( 2 , 3 ) , ( 3 , 5 ) , ( 4 , 6 ) , ( 5 , 1 ) , ( 6 , 4 ) } , 𝑔 is the inverse of 𝑓
  • D β„Ž = { ( 1 , 1 ) , ( 2 , 3 ) , ( 3 , 3 ) , ( 4 , 6 ) , ( 5 , 5 ) , ( 6 , 4 ) } , 𝑔 is the inverse of β„Ž
  • E β„Ž = { ( 2 , 1 ) , ( 3 , 2 ) , ( 5 , 3 ) , ( 6 , 4 ) , ( 1 , 5 ) , ( 4 , 6 ) } , β„Ž is the inverse of 𝑓

Q6:

Let 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 1 3 π‘₯ βˆ’ 2 . Solve 𝑦 = 𝑓 ( π‘₯ ) to find an expression for 𝑓 ( π‘₯ ) βˆ’ 1 .

  • A 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 1 3 π‘₯ βˆ’ 2 βˆ’ 1
  • B 𝑓 ( π‘₯ ) = 1 βˆ’ 2 π‘₯ 3 π‘₯ βˆ’ 2 βˆ’ 1
  • C 𝑓 ( π‘₯ ) = 2 𝑦 βˆ’ 1 3 𝑦 βˆ’ 2 βˆ’ 1
  • D 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 1 2 βˆ’ 3 π‘₯ βˆ’ 1
  • E 𝑓 ( π‘₯ ) = 2 𝑦 βˆ’ 1 2 βˆ’ 3 𝑦 βˆ’ 1

Find 𝑓 ( π‘₯ ) βˆ’ 1 and 𝑔 ( π‘₯ ) βˆ’ 1 when 𝑓 ( π‘₯ ) = 5 π‘₯ + 2 2 π‘₯ βˆ’ 5 and 𝑔 ( π‘₯ ) = 5 π‘₯ + 2 2 π‘₯ + 1 .

  • A 𝑓 ( π‘₯ ) = 5 π‘₯ + 2 2 π‘₯ βˆ’ 5 βˆ’ 1 , 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 2 βˆ’ 2 π‘₯ + 5 βˆ’ 1
  • B 𝑓 ( π‘₯ ) = 2 βˆ’ 5 π‘₯ 2 π‘₯ βˆ’ 5 βˆ’ 1 , 𝑔 ( π‘₯ ) = 2 βˆ’ π‘₯ βˆ’ 2 π‘₯ + 5 βˆ’ 1
  • C 𝑓 ( π‘₯ ) = 5 𝑦 + 2 2 𝑦 βˆ’ 5 βˆ’ 1 , 𝑔 ( π‘₯ ) = 5 𝑦 + 2 2 𝑦 + 1 βˆ’ 1
  • D 𝑓 ( π‘₯ ) = 5 𝑦 + 2 2 𝑦 + 5 βˆ’ 1 , 𝑔 ( π‘₯ ) = 𝑦 βˆ’ 2 βˆ’ 2 𝑦 βˆ’ 5 βˆ’ 1
  • E 𝑓 ( π‘₯ ) = 5 π‘₯ + 2 2 π‘₯ + 5 βˆ’ 1 , 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 2 βˆ’ 2 π‘₯ βˆ’ 5 βˆ’ 1

Let 𝑓 ( π‘₯ ) = π‘Ž π‘₯ + 𝑏 𝑐 π‘₯ + 𝑑 and 𝑔 ( π‘₯ ) = 𝑑 π‘₯ βˆ’ 𝑏 βˆ’ 𝑐 π‘₯ + π‘Ž . By considering the simplification of 𝑓 ( 𝑔 ( π‘₯ ) ) , what, if any, is the relation between 𝑓 and 𝑔 ?

  • AThey are inverses of each other.
  • BThere is no relation between them.
  • CThey are the same.
  • D 𝑔 = 𝑓 2
  • EThey are negatives of each other.

Let 𝑓 ( π‘₯ ) = 2 π‘₯ + 1 3 π‘₯ βˆ’ 2 , 𝑔 ( π‘₯ ) = βˆ’ 2 π‘₯ βˆ’ 1 βˆ’ 3 π‘₯ + 2 , 𝑔 ( π‘₯ ) = 4 π‘₯ + 2 6 π‘₯ βˆ’ 4 1 2 a n d . By considering the simplifications of 𝑓 βˆ’ 𝑔 1 and 𝑓 βˆ’ 𝑔 2 , what, if any, are the relations between these functions?

  • AThey are all the same.
  • B 𝑔 1 is the inverse of 𝑓 .
  • CThere is no relation between them.
  • D 𝑔 1 is the negative of 𝑓 , and 𝑔 2 is twice 𝑓 .

Suppose that 𝑓 = 𝑓 βˆ’ 1 , where 𝑓 ( π‘₯ ) = π‘Ž π‘₯ + 𝑏 𝑐 π‘₯ + 𝑑 . What must be true of π‘Ž , 𝑏 , 𝑐 , and 𝑑 ?

  • Ano conditions on any of the numbers
  • B 𝑑 = βˆ’ π‘Ž , no conditions on 𝑐 and 𝑏
  • C π‘Ž = 2 , 𝑏 = βˆ’ 1 , 𝑐 = 3 , 𝑑 = βˆ’ 2 a n d

Q7:

Which of the following is a necessary condition for a function 𝑓 to be invertible?

  • A 𝑓 has to be both one-to-one and onto.
  • B 𝑓 has to be onto.
  • C 𝑓 has to be a function.
  • D 𝑓 has to be one-to-one.

Q8:

Which of the following functions is NOT its own inverse?

  • A 𝑓 ( π‘₯ ) = 1 π‘₯ + 6 + 6
  • B 𝑓 ( π‘₯ ) = 6 βˆ’ π‘₯
  • C 𝑓 ( π‘₯ ) = 6 π‘₯
  • D 𝑓 ( π‘₯ ) = βˆ’ 6 π‘₯
  • E 𝑓 ( π‘₯ ) = π‘₯
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