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.

  • A𝑓(𝑥)=3(𝑥−2)
  • B𝑓(𝑥)=13(𝑥+2)
  • C𝑓(𝑥)=13(𝑥−2)
  • D𝑓(𝑥)=2(𝑥+3)

Q2:

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

  • A𝑓(𝑥)=4𝑥
  • B𝑓(𝑥)=𝑥4
  • C𝑓(𝑥)=−𝑥4
  • D𝑓(𝑥)=−4𝑥
  • E𝑓(𝑥)=−4𝑥

Q3:

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

  • A𝑓(𝑥)=2−𝑥
  • B𝑓(𝑥)=√2−𝑥
  • C𝑓(𝑥)=2+𝑥
  • D𝑓(𝑥)=𝑥−2
  • E𝑓(𝑥)=(2−𝑥)

Q4:

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

  • A𝑓(𝑥)=6−√𝑥+5
  • B𝑓(𝑥)=√𝑥−5+6
  • C𝑓(𝑥)=√𝑥+5−6
  • D𝑓(𝑥)=−6−√𝑥+5
  • E𝑓(𝑥)=√𝑥−6+5

Q5:

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

  • A𝑓={(2,−7),(−2,−4),(−6,−5),(−10,−2)}
  • B𝑓=12,17,−12,14,−16,15,−110,12
  • C𝑓=2,17,−2,14,−6,15,−10,12
  • D𝑓={(7,2),(4,−2),(5,−6),(2,−10)}
  • E𝑓={(−10,2),(−6,5),(−2,4),(2,7)}

Q6:

Solve √𝑥−7=−3.

  • A𝑥=4
  • B𝑥=16
  • C𝑥=2
  • D𝑥=10
  • 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?

  • A𝑥+7
  • B𝑐+7
  • C𝑥+7
  • D𝑥−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.

  • A𝑓(𝑥)=𝑥6
  • B𝑓(𝑥)=−6𝑥
  • C𝑓(𝑥)=𝑥6
  • D𝑓(𝑥)=6√𝑥
  • E𝑓(𝑥)=16𝑥

Q10:

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

  • A7𝑦=5𝑥
  • B𝑦=𝑥−57
  • C𝑦=−7𝑥−5
  • D𝑦=−7𝑥+5
  • E𝑦=𝑥+57

Q11:

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

  • A[0,7]
  • B(−∞,0] or [0,∞)
  • C(−∞,0]
  • Dℝ
  • 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.

  • A𝑓(𝑥)=(𝑥−3)+2 where 𝑥⩾3
  • B𝑓(𝑥)=12+√𝑥+3 where 𝑥⩾3
  • C𝑓(𝑥)=(𝑥+2)+3 where 𝑥⩾2
  • D𝑓(𝑥)=(𝑥−2)−3 where 𝑥⩾2
  • E𝑓(𝑥)=−2+√𝑥−3 where 𝑥⩾3

Q14:

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

  • A𝑓(𝑥)=√𝑥−3−2
  • B𝑓(𝑥)=√𝑥+2+3
  • C𝑓(𝑥)=√𝑥−2−3
  • D𝑓(𝑥)=−3−√𝑥−2
  • E𝑓(𝑥)=3−√𝑥−2

Q15:

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

  • A𝑓(𝑥)=(−𝑥−3) for 𝑥≥−3
  • B𝑓(𝑥)=−(𝑥−3) for 𝑥≥3
  • C𝑓(𝑥)=(𝑥−3) for 𝑥≥3
  • D𝑓(𝑥)=(𝑥−2) for 𝑥≥2

Q16:

Find 𝑓(𝑥) for 𝑓(𝑥)=3+√𝑥.

  • A𝑓(𝑥)=𝑥−3
  • B𝑓(𝑥)=(𝑥−3)
  • C𝑓(𝑥)=−(𝑥−3)
  • D𝑓(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.

  • A𝑙=32.2𝑇2𝜋, 3.26 feet
  • B𝑙=2𝜋𝑇32.2, 0.15 feet
  • C𝑙=32.2𝑇2𝜋, 20.5 feet
  • D𝑙=32.2𝑇2𝜋, 10.2 feet
  • E𝑙=2𝜋𝑇32.2, 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?

  • A𝑐>7
  • B𝑐<7
  • Cany 𝑐≥0
  • D𝑐<0
  • E𝑐≥−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?

  • A𝑥>0
  • B𝑥≤−3
  • C𝑥≥−3
  • D𝑥≥0
  • E𝑥<0

What is the range of the inverse?

  • A𝑥>0
  • B𝑥≥−3
  • C𝑥≥0
  • D𝑥≤−3
  • E𝑥<−3

Give a formula for the inverse.

  • A𝑓(𝑥)=𝑥3−3
  • B𝑓(𝑥)=−𝑥3+3
  • C𝑓(𝑥)=−𝑥3−3
  • D𝑓(𝑥)=𝑥3+3
  • E𝑓(𝑥)=3𝑥−13

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.

𝑥1234𝑑6
𝑓(𝑥)−31𝑎−19−10114
𝑥−31−26−19−101𝑒
𝑔(𝑥)12𝑏𝑐56
  • A𝑎=−26, 𝑏=3, 𝑐=4, 𝑑=5, 𝑒=14
  • B𝑎=2, 𝑏=−19, 𝑐=4, 𝑑=5, 𝑒=14
  • C𝑎=2, 𝑏=−19, 𝑐=4, 𝑑=1, 𝑒=6
  • D𝑎=−26, 𝑏=−19, 𝑐=4, 𝑑=1, 𝑒=14
  • E𝑎=−26, 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).

  • A𝑡(𝑑)=𝑑50, 3.6 hours
  • B𝑡(𝑑)=502𝑑, 0.13889 hours
  • C𝑡(𝑑)=𝑑50, 0.2778 hours
  • D𝑡(𝑑)=2𝑑50, 7.2 hours
  • E𝑡(𝑑)=50𝑑, 0.2778 hours

Q23:

Use the table to find ℎ(30).

𝑥15304560
ℎ(𝑥)20253035

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?

  • A𝑓(𝑥)=1𝑥, 𝑔(𝑥)=1𝑥
  • B𝑓(𝑥)=𝑥, 𝑔(𝑥)=−𝑥
  • C𝑓(𝑥)=1𝑥, 𝑔(𝑥)=𝑥
  • D𝑓(𝑥)=2𝑥–3, 𝑔(𝑥)=3𝑥+2

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