Worksheet: Inverse of a Function

Practice

In this worksheet, we will practice finding the inverse of a function by changing the subject of the formula.

Q1:

Determine the inverse of 𝑓 ( 𝑥 ) = 1 3 𝑥 + 2 .

  • A 𝑓 ( 𝑥 ) = 2 ( 𝑥 + 3 )
  • B 𝑓 ( 𝑥 ) = 1 3 ( 𝑥 + 2 )
  • C 𝑓 ( 𝑥 ) = 1 3 ( 𝑥 − 2 )
  • D 𝑓 ( 𝑥 ) = 3 ( 𝑥 − 2 )

Q2:

Find the inverse of the function 𝑓 ( 𝑥 ) = 4 𝑥 .

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

Q3:

Find the inverse of the function 𝑓 ( 𝑥 ) = √ 2 − 𝑥 .

  • A 𝑓 ( 𝑥 ) = 𝑥 − 2
  • B 𝑓 ( 𝑥 ) = 2 + 𝑥
  • C 𝑓 ( 𝑥 ) = √ 2 − 𝑥
  • D 𝑓 ( 𝑥 ) = 2 − 𝑥
  • E 𝑓 ( 𝑥 ) = ( 2 − 𝑥 )

Q4:

Determine the inverse function of 𝑓 ( 𝑥 ) = ( 𝑥 + 6 ) − 5 , where 𝑥 ≥ − 6 .

  • A 𝑓 ( 𝑥 ) = √ 𝑥 − 6 + 5
  • B 𝑓 ( 𝑥 ) = √ 𝑥 − 5 + 6
  • C 𝑓 ( 𝑥 ) = 6 − √ 𝑥 + 5
  • D 𝑓 ( 𝑥 ) = √ 𝑥 + 5 − 6
  • E 𝑓 ( 𝑥 ) = − 6 − √ 𝑥 + 5

Q5:

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

  • A 𝑓 = 2 , 1 7 , − 2 , 1 4 , − 6 , 1 5 , − 1 0 , 1 2
  • B 𝑓 = { ( 2 , − 7 ) , ( − 2 , − 4 ) , ( − 6 , − 5 ) , ( − 1 0 , − 2 ) }
  • C 𝑓 = 1 2 , 1 7 , − 1 2 , 1 4 , − 1 6 , 1 5 , − 1 1 0 , 1 2
  • D 𝑓 = { ( 7 , 2 ) , ( 4 , − 2 ) , ( 5 , − 6 ) , ( 2 , − 1 0 ) }
  • E 𝑓 = { ( − 1 0 , 2 ) , ( − 6 , 5 ) , ( − 2 , 4 ) , ( 2 , 7 ) }

Q6:

Solve √ 𝑥 − 7 = − 3 .

  • A 𝑥 = 2
  • B 𝑥 = 1 0
  • C 𝑥 = 1 6
  • D This has no solution.
  • E 𝑥 = 4

Q7:

Bassem is trying to find the inverse of 𝑓 ( 𝑥 ) = √ 𝑥 − 7 . He sets 𝑐 = √ 𝑥 − 7 and then finds 𝑐 = 𝑥 − 7 and then 𝑥 = 7 + 𝑐 . What does Bassem determine 𝑓 ( 𝑥 ) to be?

  • A ( 𝑥 − 7 )
  • B 𝑐 + 7
  • C 𝑥 + 7
  • D 𝑥 + 7
  • E 𝑥 − 7

Q8:

Let 𝑓 ( 𝑥 ) = 3 𝑥 + 5 and 𝑔 ( 𝑥 ) = 𝑥 − 5 3 . Is it true that 𝑓 is the inverse of 𝑔 and 𝑔 is the inverse of 𝑓 ?

  • Ayes
  • Bno

Q9:

Find the inverse of the function 𝑓 ( 𝑥 ) = 6 𝑥 .

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

Q10:

What is the inverse of the function 𝑦 = 7 𝑥 − 5 ?

  • A 𝑦 = 𝑥 − 5 7
  • B 𝑦 = − 7 𝑥 + 5
  • C 7 𝑦 = 5 𝑥
  • D 𝑦 = 𝑥 + 5 7
  • E 𝑦 = − 7 𝑥 − 5

Q11:

Determine the domain on which the function 𝑓 ( 𝑥 ) = 7 𝑥 has an inverse.

  • A ] − ∞ , 0 ]
  • B ℝ
  • C [ 0 , 7 ]
  • D ] − ∞ , 0 ] or [ 0 , ∞ [
  • E { 7 }

Q12:

If 𝑓 − 1 is the inverse function of the function 𝑓 then which of the following statements is true?

  • Arange of 𝑓 = − 1 range of 𝑓
  • Bdomain of 𝑓 = − 1 domain of 𝑓
  • Crange of 𝑓 = − 1 domain of 𝑓 − 1
  • Ddomain of 𝑓 = − 1 range of 𝑓
  • Edomain of 𝑓 = ℝ − − 1 range of 𝑓

Q13:

Find the inverse of the function 𝑓 ( 𝑥 ) = 2 + √ 𝑥 + 3 .

  • A 𝑓 ( 𝑥 ) = − 2 + √ 𝑥 − 3 where 𝑥 ⩾ 3
  • B 𝑓 ( 𝑥 ) = ( 𝑥 + 2 ) + 3 where 𝑥 ⩾ 2
  • C 𝑓 ( 𝑥 ) = 1 2 + √ 𝑥 + 3 where 𝑥 ⩾ 3
  • D 𝑓 ( 𝑥 ) = ( 𝑥 − 2 ) − 3 where 𝑥 ⩾ 2
  • E 𝑓 ( 𝑥 ) = ( 𝑥 − 3 ) + 2 where 𝑥 ⩾ 3

Q14:

Find the inverse of the function 𝑓 ( 𝑥 ) = 𝑥 + 6 𝑥 + 1 1 , where 𝑥 ≥ − 3 .

  • A 𝑓 ( 𝑥 ) = √ 𝑥 − 3 − 2
  • B 𝑓 ( 𝑥 ) = √ 𝑥 + 2 + 3
  • C 𝑓 ( 𝑥 ) = 3 − √ 𝑥 − 2
  • D 𝑓 ( 𝑥 ) = √ 𝑥 − 2 − 3
  • E 𝑓 ( 𝑥 ) = − 3 − √ 𝑥 − 2

Q15:

Find 𝑓 ( 𝑥 ) − 1 for 𝑓 ( 𝑥 ) = √ 𝑥 + 3 and state the domain.

  • A 𝑓 ( 𝑥 ) = ( − 𝑥 − 3 ) − 1 2 for 𝑥 ≥ − 3
  • B 𝑓 ( 𝑥 ) = − ( 𝑥 − 3 ) − 1 2 for 𝑥 ≥ 3
  • C 𝑓 ( 𝑥 ) = ( 𝑥 − 2 ) − 1 3 for 𝑥 ≥ 2
  • D 𝑓 ( 𝑥 ) = ( 𝑥 − 3 ) − 1 2 for 𝑥 ≥ 3

Q16:

Find 𝑓 ( 𝑥 ) − 1 for 𝑓 ( 𝑥 ) = 3 + √ 𝑥 3 .

  • A 𝑓 ( 𝑥 ) = − ( 𝑥 − 3 ) − 1 3
  • B 𝑓 ( 𝑥 ) = 𝑥 − 3 − 1
  • C 𝑓 ( 𝑥 ) = − 3 − √ 𝑥 − 1 3
  • D 𝑓 ( 𝑥 ) = ( 𝑥 − 3 ) − 1 3

Q17:

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 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 , 10.2 feet
  • B 𝑙 = 3 2 . 2 𝑇 2 𝜋 2 , 20.5 feet
  • C 𝑙 = 2 𝜋 𝑇 3 2 . 2 2 , 0.15 feet
  • D 𝑙 = 3 2 . 2 𝑇 2 𝜋 2 , 3.26 feet
  • E 𝑙 = 2 𝜋 𝑇 3 2 . 2 , 0.39 feet

Q18:

The figure below represents the function 𝑓 𝑋 → 𝑌 : . Find the value of 𝑓 ( 4 ) .

Q19:

For what numbers 𝑐 can we solve √ 𝑥 − 7 = 𝑐 ?

  • A 𝑐 > 7
  • B 𝑐 < 0
  • C 𝑐 < 7
  • D any 𝑐 ≥ 0
  • E 𝑐 ≥ − 7

Q20:

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

What is the domain of the inverse?

  • A 𝑥 > 0
  • B 𝑥 ≥ − 3
  • C 𝑥 < 0
  • D 𝑥 ≥ 0
  • E 𝑥 ≤ − 3

What is the range of the inverse?

  • A 𝑥 ≤ − 3
  • B 𝑥 < − 3
  • C 𝑥 > 0
  • D 𝑥 ≥ 0
  • E 𝑥 ≥ − 3

Give a formula for the inverse.

  • A 𝑓 ( 𝑥 ) = 3 𝑥 − 1 3 − 1
  • B 𝑓 ( 𝑥 ) = 𝑥 3 − 3 − 1
  • C 𝑓 ( 𝑥 ) = − 𝑥 3 − 3 − 1
  • D 𝑓 ( 𝑥 ) = − 𝑥 3 + 3 − 1
  • E 𝑓 ( 𝑥 ) = 𝑥 3 + 3 − 1

Q21:

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

𝑥 1 2 3 4 𝑑 6
𝑓 ( 𝑥 ) − 3 1 𝑎 − 1 9 − 1 0 1 14
𝑥 − 3 1 − 2 6 − 1 9 − 1 0 1 𝑒
𝑔 ( 𝑥 ) 1 2 b c 5 6
  • A 𝑎 = 2 , 𝑏 = − 1 9 , 𝑐 = 4 , 𝑑 = 5 , 𝑒 = 1 4
  • B 𝑎 = − 2 6 , 𝑏 = 3 , 𝑐 = 4 , 𝑑 = 1 , 𝑒 = 6
  • C 𝑎 = 2 , 𝑏 = − 1 9 , 𝑐 = 4 , 𝑑 = 1 , 𝑒 = 6
  • D 𝑎 = − 2 6 , 𝑏 = 3 , 𝑐 = 4 , 𝑑 = 5 , 𝑒 = 1 4
  • E 𝑎 = − 2 6 , 𝑏 = − 1 9 , 𝑐 = 4 , 𝑑 = 1 , 𝑒 = 1 4

Q22:

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

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

Q23:

Use the table to find ℎ ( 3 0 ) − 1 .

𝑥 15 30 45 60
ℎ ( 𝑥 ) 20 25 30 35

Q24:

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

  • Ayes
  • Bno

Q25:

Which of the following pairs of functions are inverses for all 𝑥 ∈ ℝ − 0 ?

  • A 𝑓 ( 𝑥 ) = 1 𝑥 , 𝑔 ( 𝑥 ) = 𝑥
  • B 𝑓 ( 𝑥 ) = 2 𝑥 – 3 , 𝑔 ( 𝑥 ) = 3 𝑥 + 2
  • C 𝑓 ( 𝑥 ) = 𝑥 , 𝑔 ( 𝑥 ) = − 𝑥
  • D 𝑓 ( 𝑥 ) = 1 𝑥 , 𝑔 ( 𝑥 ) = 1 𝑥

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