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Worksheet: Inverse of a Function

Q1:

Find the inverse of the function 𝑓 ( π‘₯ ) = 4 π‘₯ .

  • A 𝑓 ( π‘₯ ) = 4 π‘₯ βˆ’ 1
  • B 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ βˆ’ 1
  • C 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ βˆ’ 1
  • D 𝑓 ( π‘₯ ) = π‘₯ 4 βˆ’ 1
  • E 𝑓 ( π‘₯ ) = βˆ’ π‘₯ 4 βˆ’ 1

Q2:

Find the inverse of the function 𝑓 ( π‘₯ ) = 1 0 π‘₯ .

  • A 𝑓 ( π‘₯ ) = 1 0 π‘₯ βˆ’ 1
  • B 𝑓 ( π‘₯ ) = βˆ’ 1 0 π‘₯ βˆ’ 1
  • C 𝑓 ( π‘₯ ) = βˆ’ 1 0 π‘₯ βˆ’ 1
  • D 𝑓 ( π‘₯ ) = π‘₯ 1 0 βˆ’ 1
  • E 𝑓 ( π‘₯ ) = βˆ’ π‘₯ 1 0 βˆ’ 1

Q3:

Determine the inverse of 𝑓 ( π‘₯ ) = 1 3 π‘₯ + 2 .

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

Q4:

Determine the inverse of 𝑓 ( π‘₯ ) = 1 5 π‘₯ + 3 .

  • A 𝑓 ( π‘₯ ) = 3 ( π‘₯ + 5 ) βˆ’ 1
  • B 𝑓 ( π‘₯ ) = 1 5 ( π‘₯ + 3 ) βˆ’ 1
  • C 𝑓 ( π‘₯ ) = 1 5 ( π‘₯ βˆ’ 3 ) βˆ’ 1
  • D 𝑓 ( π‘₯ ) = 5 ( π‘₯ βˆ’ 3 ) βˆ’ 1

Q5:

Find the inverse of the function .

  • A
  • B
  • C
  • D
  • E

Q6:

Find the inverse of the function .

  • A
  • B
  • C
  • D
  • E

Q7:

Determine the inverse function of 𝑓 ( π‘₯ ) = ( π‘₯ + 6 ) βˆ’ 5 2 , where π‘₯ β‰₯ βˆ’ 6 .

  • A 𝑓 ( π‘₯ ) = √ π‘₯ βˆ’ 6 + 5 βˆ’ 1
  • B 𝑓 ( π‘₯ ) = √ π‘₯ βˆ’ 5 + 6 βˆ’ 1
  • C 𝑓 ( π‘₯ ) = 6 βˆ’ √ π‘₯ + 5 βˆ’ 1
  • D 𝑓 ( π‘₯ ) = √ π‘₯ + 5 βˆ’ 6 βˆ’ 1
  • E 𝑓 ( π‘₯ ) = βˆ’ 6 βˆ’ √ π‘₯ + 5 βˆ’ 1

Q8:

Find the inverse of the function 𝑓 ( π‘₯ ) = 6 π‘₯ 3 .

  • A 𝑓 ( π‘₯ ) = βˆ’ 6 π‘₯ βˆ’ 1 3
  • B 𝑓 ( π‘₯ ) = 6 √ π‘₯ βˆ’ 1 3
  • C 𝑓 ( π‘₯ ) = 1 6 π‘₯ βˆ’ 1 3
  • D 𝑓 ( π‘₯ ) = ο„ž π‘₯ 6 βˆ’ 1 3
  • E 𝑓 ( π‘₯ ) = π‘₯ 6 βˆ’ 1 3

Q9:

Find 𝑓 ( π‘₯ ) βˆ’ 1 for 𝑓 ( π‘₯ ) = √ π‘₯ + 3 and state the domain.

  • A 𝑓 ( π‘₯ ) = ( βˆ’ π‘₯ βˆ’ 3 ) βˆ’ 1 2 for π‘₯ β‰₯ βˆ’ 3
  • B 𝑓 ( π‘₯ ) = βˆ’ ( π‘₯ βˆ’ 3 ) βˆ’ 1 2 for π‘₯ β‰₯ 3
  • C 𝑓 ( π‘₯ ) = ( π‘₯ βˆ’ 2 ) βˆ’ 1 3 for π‘₯ β‰₯ 2
  • D 𝑓 ( π‘₯ ) = ( π‘₯ βˆ’ 3 ) βˆ’ 1 2 for π‘₯ β‰₯ 3

Q10:

Find 𝑓 ( π‘₯ ) βˆ’ 1 for 𝑓 ( π‘₯ ) = 3 + √ π‘₯ 3 .

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

Q11:

The following tables are partially filled for functions 𝑓 and 𝑔 that are inverses of each other. Determine the values of π‘Ž , 𝑏 , 𝑐 , 𝑑 , and 𝑒 .

π‘₯ 1 2 3 4 𝑑 6
𝑓 ( π‘₯ ) βˆ’ 3 1 π‘Ž βˆ’ 1 9 βˆ’ 1 0 1 14
π‘₯ βˆ’ 3 1 βˆ’ 2 6 βˆ’ 1 9 βˆ’ 1 0 1 𝑒
𝑔 ( π‘₯ ) 1 2 b c 5 6
  • A π‘Ž = 2 , 𝑏 = βˆ’ 1 9 , 𝑐 = 4 , 𝑑 = 5 , 𝑒 = 1 4
  • B π‘Ž = βˆ’ 2 6 , 𝑏 = 3 , 𝑐 = 4 , 𝑑 = 1 , 𝑒 = 6
  • C π‘Ž = 2 , 𝑏 = βˆ’ 1 9 , 𝑐 = 4 , 𝑑 = 1 , 𝑒 = 6
  • D π‘Ž = βˆ’ 2 6 , 𝑏 = 3 , 𝑐 = 4 , 𝑑 = 5 , 𝑒 = 1 4
  • E π‘Ž = βˆ’ 2 6 , 𝑏 = βˆ’ 1 9 , 𝑐 = 4 , 𝑑 = 1 , 𝑒 = 1 4

Q12:

A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, 𝑑 , given in hours by 𝑑 ( 𝑑 ) = 5 0 𝑑 . Find the inverse function expressing the time of travel in terms of the distance traveled. Find 𝑑 ( 1 8 0 ) .

  • A 𝑑 ( 𝑑 ) = 2 𝑑 5 0 , 7.2 hours
  • B 𝑑 ( 𝑑 ) = 5 0 𝑑 , 0.2778 hours
  • C 𝑑 ( 𝑑 ) = 5 0 2 𝑑 , 0.13889 hours
  • D 𝑑 ( 𝑑 ) = 𝑑 5 0 , 3.6 hours
  • E 𝑑 ( 𝑑 ) = 𝑑 5 0 , 0.2778 hours

Q13:

Use the table to find β„Ž ( 3 0 ) βˆ’ 1 .

π‘₯ 15 30 45 60
β„Ž ( π‘₯ ) 20 25 30 35

Q14:

The solid part of the following graph of 𝑓 ( π‘₯ ) = | 3 ( π‘₯ + 3 ) | shows how we can restrict the domain to obtain an inverse.

What is the domain of the inverse?

  • A π‘₯ > 0
  • B π‘₯ β‰₯ βˆ’ 3
  • C π‘₯ < 0
  • D π‘₯ β‰₯ 0
  • E π‘₯ ≀ βˆ’ 3

What is the range of the inverse?

  • A π‘₯ ≀ βˆ’ 3
  • B π‘₯ < βˆ’ 3
  • C π‘₯ > 0
  • D π‘₯ β‰₯ 0
  • E π‘₯ β‰₯ βˆ’ 3

Give a formula for the inverse.

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

Q15:

Jacob is trying to find the inverse of 𝑓 ( π‘₯ ) = √ π‘₯ βˆ’ 7 . He sets 𝑐 = √ π‘₯ βˆ’ 7 and then finds 𝑐 = π‘₯ βˆ’ 7 2 and then π‘₯ = 7 + 𝑐 2 . What does Tom determine 𝑓 ( π‘₯ ) βˆ’ 1 to be?

  • A ( π‘₯ βˆ’ 7 ) 2
  • B 𝑐 + 7 2
  • C π‘₯ + 7
  • D π‘₯ + 7 2
  • E π‘₯ βˆ’ 7

Q16:

Does the function 𝑓 , where 𝑓 = { ( 5 , 3 ) , ( 9 , 7 ) , ( 1 1 , 1 0 ) } , have an inverse?

  • Ayes
  • Bno

Q17:

Which of the following pairs of functions are inverses for all π‘₯ ∈ ℝ βˆ’ 0 ?

  • A 𝑓 ( π‘₯ ) = 1 π‘₯ , 𝑔 ( π‘₯ ) = π‘₯
  • B 𝑓 ( π‘₯ ) = 2 π‘₯ – 3 , 𝑔 ( π‘₯ ) = 3 π‘₯ + 2
  • C 𝑓 ( π‘₯ ) = π‘₯ , 𝑔 ( π‘₯ ) = βˆ’ π‘₯
  • D 𝑓 ( π‘₯ ) = 1 π‘₯ , 𝑔 ( π‘₯ ) = 1 π‘₯

Q18:

Which of the following functions does not have an inverse over its whole domain?

  • A 𝑓 ( π‘₯ ) = 1 π‘₯
  • B 𝑓 ( π‘₯ ) = 2 
  • C 𝑓 ( π‘₯ ) = 2 π‘₯
  • D 𝑓 ( π‘₯ ) = π‘₯ 

Q19:

Solve √ π‘₯ βˆ’ 7 = βˆ’ 3 .

  • A π‘₯ = 2
  • B π‘₯ = 1 0
  • C π‘₯ = 1 6
  • D This has no solution.
  • E π‘₯ = 4

Q20:

The period 𝑇 , in seconds, of a simple pendulum as a function of its length 𝑙 , in feet, is given by 𝑇 ( 𝑙 ) = 2 πœ‹ ο„ž 𝑙 3 2 . 2 . Express 𝑙 as a function of 𝑇 , and determine the length of a pendulum with a period of 2 seconds.

  • A 𝑙 = 3 2 . 2 𝑇 2 πœ‹ , 10.2 feet
  • B 𝑙 = 3 2 . 2 𝑇 2 πœ‹ 2 , 20.5 feet
  • C 𝑙 = ο€Ό 2 πœ‹ 𝑇 3 2 . 2  2 , 0.15 feet
  • D 𝑙 = 3 2 . 2 ο€Ό 𝑇 2 πœ‹  2 , 3.26 feet
  • E 𝑙 = 2 πœ‹ 𝑇 3 2 . 2 , 0.39 feet

Q21:

Find the inverse of the function , where .

  • A
  • B
  • C
  • D
  • E

Q22:

For what numbers 𝑐 can we solve √ π‘₯ βˆ’ 7 = 𝑐 ?

  • A 𝑐 > 7
  • B 𝑐 < 0
  • C 𝑐 < 7
  • D any 𝑐 β‰₯ 0
  • E 𝑐 β‰₯ βˆ’ 7

Q23:

The figure below represents the function 𝑓 𝑋 β†’ π‘Œ : . Find the value of 𝑓 ( 4 ) βˆ’ 1 .

Q24:

Find the inverse of the function 𝑓 = { ( 2 , 7 ) , ( βˆ’ 2 , 4 ) , ( βˆ’ 6 , 5 ) , ( βˆ’ 1 0 , 2 ) } .

  • A 𝑓 =  ο€Ό 2 , 1 7  , ο€Ό βˆ’ 2 , 1 4  , ο€Ό βˆ’ 6 , 1 5  , ο€Ό βˆ’ 1 0 , 1 2   βˆ’ 1
  • B 𝑓 = { ( 2 , βˆ’ 7 ) , ( βˆ’ 2 , βˆ’ 4 ) , ( βˆ’ 6 , βˆ’ 5 ) , ( βˆ’ 1 0 , βˆ’ 2 ) } βˆ’ 1
  • C 𝑓 =  ο€Ό 1 2 , 1 7  , ο€Ό βˆ’ 1 2 , 1 4  , ο€Ό βˆ’ 1 6 , 1 5  , ο€Ό βˆ’ 1 1 0 , 1 2   βˆ’ 1
  • D 𝑓 = { ( 7 , 2 ) , ( 4 , βˆ’ 2 ) , ( 5 , βˆ’ 6 ) , ( 2 , βˆ’ 1 0 ) } βˆ’ 1
  • E 𝑓 = { ( βˆ’ 1 0 , 2 ) , ( βˆ’ 6 , 5 ) , ( βˆ’ 2 , 4 ) , ( 2 , 7 ) } βˆ’ 1

Q25:

Find the inverse of the function .

  • A where
  • B where
  • C where
  • D where
  • E where