Worksheet: Inverse of a Function

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)=13𝑥+2.

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

Q2:

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

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

Q3:

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

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

Q4:

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

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

Q5:

Find the inverse of the function 𝑓={(2,7),(−2,4),(−6,5),(−10,2)}.

  • A 𝑓 = { ( 2 , − 7 ) , ( − 2 , − 4 ) , ( − 6 , − 5 ) , ( − 1 0 , − 2 ) }
  • B 𝑓 = 1 2 , 1 7 , − 1 2 , 1 4 , − 1 6 , 1 5 , − 1 1 0 , 1 2
  • C 𝑓 = 2 , 1 7 , − 2 , 1 4 , − 6 , 1 5 , − 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 = 4
  • B 𝑥 = 1 6
  • C 𝑥 = 2
  • D 𝑥 = 1 0
  • EThis has no solution.

Q7:

Jacob 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 Jacob determine 𝑓(𝑥) to be?

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

Q8:

Let 𝑓(𝑥)=3𝑥+5 and 𝑔(𝑥)=𝑥−53. 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)=6𝑥.

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

Q10:

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

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

Q11:

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

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

Q12:

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

  • Arange of 𝑓= domain of 𝑓
  • Bdomain of 𝑓= range of 𝑓
  • Cdomain of 𝑓= domain of 𝑓
  • Drange of 𝑓= range of 𝑓
  • Edomain of 𝑓=ℝ− range of 𝑓

Q13:

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

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

Q14:

Find the inverse of the function 𝑓(𝑥)=𝑥+6𝑥+11, where 𝑥≥−3.

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

Q15:

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

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

Q16:

Find 𝑓(𝑥) for 𝑓(𝑥)=3+√𝑥.

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

Q17:

The period 𝑇, in seconds, of a simple pendulum as a function of its length 𝑙, in feet, is given by 𝑇(𝑙)=2𝜋𝑙32.2. 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 , 3.26 feet
  • B 𝑙 = 2 𝜋 𝑇 3 2 . 2 , 0.15 feet
  • C 𝑙 = 3 2 . 2 𝑇 2 𝜋 , 20.5 feet
  • D 𝑙 = 3 2 . 2 𝑇 2 𝜋 , 10.2 feet
  • E 𝑙 = 2 𝜋 𝑇 3 2 . 2 , 0.39 feet

Q18:

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

  • A10
  • B13
  • C8
  • D4

Q19:

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

  • A 𝑐 > 7
  • B 𝑐 < 7
  • Cany 𝑐≥0
  • D 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 ≥ − 3
  • D 𝑥 ≥ 0
  • E 𝑥 < 0

What is the range of the inverse?

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

Give a formula for the inverse.

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

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 6 , 𝑏 = 3 , 𝑐 = 4 , 𝑑 = 5 , 𝑒 = 1 4
  • B 𝑎 = 2 , 𝑏 = − 1 9 , 𝑐 = 4 , 𝑑 = 5 , 𝑒 = 1 4
  • C 𝑎 = 2 , 𝑏 = − 1 9 , 𝑐 = 4 , 𝑑 = 1 , 𝑒 = 6
  • D 𝑎 = − 2 6 , 𝑏 = − 1 9 , 𝑐 = 4 , 𝑑 = 1 , 𝑒 = 1 4
  • E 𝑎 = − 2 6 , 𝑏 = 3 , 𝑐 = 4 , 𝑑 = 1 , 𝑒 = 6

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)=50𝑡. Find the inverse function expressing the time of travel in terms of the distance traveled. Find 𝑡(180).

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

Q23:

Use the table to find ℎ(30).

𝑥 15 30 45 60
ℎ ( 𝑥 ) 20 25 30 35

Q24:

Does the function 𝑓, where 𝑓={(5,3),(9,7),(11,10)}, 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 ) = 1 𝑥
  • B 𝑓 ( 𝑥 ) = 𝑥 , 𝑔 ( 𝑥 ) = − 𝑥
  • C 𝑓 ( 𝑥 ) = 1 𝑥 , 𝑔 ( 𝑥 ) = 𝑥
  • D 𝑓 ( 𝑥 ) = 2 𝑥 – 3 , 𝑔 ( 𝑥 ) = 3 𝑥 + 2

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