Worksheet: Inverse and Composite Functions

In this worksheet, we will practice using compositions of functions to find and justify inverses of functions.

Q1:

Which of the following functions is NOT its own inverse?

  • A 𝑓 ( 𝑥 ) = 𝑥
  • B 𝑓 ( 𝑥 ) = 4 𝑥
  • C 𝑓 ( 𝑥 ) = 8 𝑥
  • D 𝑓 ( 𝑥 ) = 8 𝑥
  • E 𝑓 ( 𝑥 ) = 8 𝑥

Q2:

Suppose 𝑔 ( 𝑥 ) = 𝑎 𝑥 + 𝑏 is the inverse function to 𝑓 ( 𝑥 ) = 2 3 𝑥 . Determine 𝑎 and 𝑏 by considering the composition 𝑓 ( 𝑔 ( 𝑥 ) ) .

  • A 𝑎 = 2 3 , 𝑏 = 1 3
  • B 𝑎 = 1 3 , 𝑏 = 2 3
  • C 𝑎 = 2 3 , 𝑏 = 1 3
  • D 𝑎 = 1 3 , 𝑏 = 2 3
  • E 𝑎 = 1 3 , 𝑏 = 2 3

Q3:

Let 𝑓 ( 𝑥 ) = 2 𝑥 1 3 𝑥 2 . Determine which of the following functions 𝑔 is the inverse of 𝑓 by checking whether 𝑓 ( 𝑔 ( 𝑥 ) ) = 𝑥 .

  • A 𝑔 ( 𝑥 ) = 𝑥 1 3 𝑥 2
  • B 𝑔 ( 𝑥 ) = 2 𝑥 1 6 𝑥 4
  • C 𝑔 ( 𝑥 ) = 2 𝑥 1 3 𝑥 2

Q4:

Determine 𝑎 and 𝑏 so that 𝑔 ( 𝑥 ) = 𝑎 + 𝑏 𝑥 is an inverse function to 𝑓 ( 𝑥 ) = ( 𝑥 3 ) by considering 𝑓 ( 𝑔 ( 𝑥 ) ) .

  • A 𝑎 = 3 , 𝑏 = 1
  • B 𝑎 = 3 , 𝑏 = 1
  • C 𝑎 = 3 , 𝑏 = 1 and 𝑎 = 3 , 𝑏 = 1
  • D 𝑎 = 3 , 𝑏 = 1 and 𝑎 = 3 , 𝑏 = 1
  • E 𝑎 = 3 , 𝑏 = 1 and 𝑎 = 3 , 𝑏 = 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 ( 𝑥 ) = 𝑓 ( 𝑔 ( 𝑥 ) ) as a list of ordered pairs. What can you conclude?

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

Q6:

Let 𝑓 ( 𝑥 ) = 2 𝑥 1 3 𝑥 2 . Solve 𝑦 = 𝑓 ( 𝑥 ) to find an expression for 𝑓 ( 𝑥 ) .

  • A 𝑓 ( 𝑥 ) = 2 𝑦 1 2 3 𝑦
  • B 𝑓 ( 𝑥 ) = 2 𝑦 1 3 𝑦 2
  • C 𝑓 ( 𝑥 ) = 2 𝑥 1 2 3 𝑥
  • D 𝑓 ( 𝑥 ) = 2 𝑥 1 3 𝑥 2
  • E 𝑓 ( 𝑥 ) = 1 2 𝑥 3 𝑥 2

Find 𝑓 ( 𝑥 ) and 𝑔 ( 𝑥 ) when 𝑓 ( 𝑥 ) = 5 𝑥 + 2 2 𝑥 5 and 𝑔 ( 𝑥 ) = 5 𝑥 + 2 2 𝑥 + 1 .

  • A 𝑓 ( 𝑥 ) = 5 𝑥 + 2 2 𝑥 5 , 𝑔 ( 𝑥 ) = 𝑥 2 2 𝑥 + 5
  • B 𝑓 ( 𝑥 ) = 5 𝑦 + 2 2 𝑦 + 5 , 𝑔 ( 𝑥 ) = 𝑦 2 2 𝑦 5
  • C 𝑓 ( 𝑥 ) = 5 𝑥 + 2 2 𝑥 + 5 , 𝑔 ( 𝑥 ) = 𝑥 2 2 𝑥 5
  • D 𝑓 ( 𝑥 ) = 5 𝑦 + 2 2 𝑦 5 , 𝑔 ( 𝑥 ) = 5 𝑦 + 2 2 𝑦 + 1
  • E 𝑓 ( 𝑥 ) = 2 5 𝑥 2 𝑥 5 , 𝑔 ( 𝑥 ) = 2 𝑥 2 𝑥 + 5

Let 𝑓 ( 𝑥 ) = 𝑎 𝑥 + 𝑏 𝑐 𝑥 + 𝑑 and 𝑔 ( 𝑥 ) = 𝑑 𝑥 𝑏 𝑐 𝑥 + 𝑎 . By considering the simplification of 𝑓 ( 𝑔 ( 𝑥 ) ) , what, if any, is the relation between 𝑓 and 𝑔 ?

  • AThere is no relation between them.
  • BThey are the same.
  • CThey are inverses of each other.
  • DThey are negatives of each other.
  • E 𝑔 = 𝑓

Let 𝑓 ( 𝑥 ) = 2 𝑥 + 1 3 𝑥 2 , 𝑔 ( 𝑥 ) = 2 𝑥 1 3 𝑥 + 2 , and 𝑔 ( 𝑥 ) = 4 𝑥 + 2 6 𝑥 4 . By considering the simplifications of 𝑓 𝑔 and 𝑓 𝑔 , what, if any, are the relations between these functions?

  • AThey are all the same.
  • BThere is no relation between them.
  • C 𝑔 is the inverse of 𝑓 .
  • D 𝑔 is the negative of 𝑓 , and 𝑔 is twice 𝑓 .

Suppose that 𝑓 = 𝑓 , where 𝑓 ( 𝑥 ) = 𝑎 𝑥 + 𝑏 𝑐 𝑥 + 𝑑 . What must be true of 𝑎 , 𝑏 , 𝑐 , and 𝑑 ?

  • A 𝑑 = 𝑎 , no conditions on 𝑐 and 𝑏
  • B 𝑎 = 2 , 𝑏 = 1 , 𝑐 = 3 , and 𝑑 = 2
  • Cno conditions on any of the numbers

Q7:

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

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

Q8:

Which of the following functions is NOT its own inverse?

  • A 𝑓 ( 𝑥 ) = 𝑥
  • B 𝑓 ( 𝑥 ) = 6 𝑥
  • C 𝑓 ( 𝑥 ) = 6 𝑥
  • D 𝑓 ( 𝑥 ) = 1 𝑥 + 6 + 6
  • E 𝑓 ( 𝑥 ) = 6 𝑥

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