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(π‘₯)=13π‘₯+2.

  • A𝑓(π‘₯)=3(π‘₯βˆ’2)
  • B𝑓(π‘₯)=13(π‘₯+2)
  • C𝑓(π‘₯)=13(π‘₯βˆ’2)
  • D𝑓(π‘₯)=2(π‘₯+3)

Q2:

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

  • A𝑓(π‘₯)=4π‘₯
  • B𝑓(π‘₯)=π‘₯4
  • C𝑓(π‘₯)=βˆ’π‘₯4
  • D𝑓(π‘₯)=βˆ’4π‘₯
  • E𝑓(π‘₯)=βˆ’4π‘₯

Q3:

Find the inverse of the function 𝑓(π‘₯)=√2βˆ’π‘₯.

  • A𝑓(π‘₯)=2βˆ’π‘₯
  • B𝑓(π‘₯)=√2βˆ’π‘₯
  • C𝑓(π‘₯)=2+π‘₯
  • D𝑓(π‘₯)=π‘₯βˆ’2
  • E𝑓(π‘₯)=(2βˆ’π‘₯)

Q4:

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

  • A𝑓(π‘₯)=6βˆ’βˆšπ‘₯+5
  • B𝑓(π‘₯)=√π‘₯βˆ’5+6
  • C𝑓(π‘₯)=√π‘₯+5βˆ’6
  • D𝑓(π‘₯)=βˆ’6βˆ’βˆšπ‘₯+5
  • E𝑓(π‘₯)=√π‘₯βˆ’6+5

Q5:

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

  • A𝑓={(2,βˆ’7),(βˆ’2,βˆ’4),(βˆ’6,βˆ’5),(βˆ’10,βˆ’2)}
  • B𝑓=12,17,ο€Όβˆ’12,14,ο€Όβˆ’16,15,ο€Όβˆ’110,12
  • C𝑓=2,17,ο€Όβˆ’2,14,ο€Όβˆ’6,15,ο€Όβˆ’10,12
  • D𝑓={(7,2),(4,βˆ’2),(5,βˆ’6),(2,βˆ’10)}
  • E𝑓={(βˆ’10,2),(βˆ’6,5),(βˆ’2,4),(2,7)}

Q6:

Solve √π‘₯βˆ’7=βˆ’3.

  • Aπ‘₯=4
  • Bπ‘₯=16
  • Cπ‘₯=2
  • Dπ‘₯=10
  • EThis has no solution.

Q7:

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 and then π‘₯=7+π‘οŠ¨. What does Jacob determine 𝑓(π‘₯) to be?

  • Aπ‘₯+7
  • B𝑐+7
  • Cπ‘₯+7
  • Dπ‘₯βˆ’7
  • E(π‘₯βˆ’7)

Q8:

Let 𝑓(π‘₯)=3π‘₯+5 and 𝑔(π‘₯)=π‘₯βˆ’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(π‘₯)=6π‘₯.

  • A𝑓(π‘₯)=ο„žπ‘₯6
  • B𝑓(π‘₯)=βˆ’6π‘₯
  • C𝑓(π‘₯)=π‘₯6
  • D𝑓(π‘₯)=6√π‘₯
  • E𝑓(π‘₯)=16π‘₯

Q10:

What is the inverse of the function 𝑦=7π‘₯βˆ’5?

  • A7𝑦=5π‘₯
  • B𝑦=π‘₯βˆ’57
  • C𝑦=βˆ’7π‘₯βˆ’5
  • D𝑦=βˆ’7π‘₯+5
  • E𝑦=π‘₯+57

Q11:

Determine the domain on which the function 𝑓(π‘₯)=7π‘₯ has an inverse.

  • A[0,7]
  • B(βˆ’βˆž,0]Β orΒ [0,∞)
  • C(βˆ’βˆž,0]
  • Dℝ
  • E{7}

Q12:

If π‘“οŠ±οŠ§ is the inverse function of the function 𝑓 then which of the following statements is true?

  • Arange of 𝑓= domain of π‘“οŠ±οŠ§
  • Bdomain of 𝑓= range of 𝑓
  • Cdomain of 𝑓= domain of 𝑓
  • Drange of 𝑓= range of 𝑓
  • Edomain of 𝑓=β„βˆ’οŠ±οŠ§ range of 𝑓

Q13:

Find the inverse of the function 𝑓(π‘₯)=2+√π‘₯+3.

  • A𝑓(π‘₯)=(π‘₯βˆ’3)+2 where π‘₯β©Ύ3
  • B𝑓(π‘₯)=12+√π‘₯+3 where π‘₯β©Ύ3
  • C𝑓(π‘₯)=(π‘₯+2)+3 where π‘₯β©Ύ2
  • D𝑓(π‘₯)=(π‘₯βˆ’2)βˆ’3 where π‘₯β©Ύ2
  • E𝑓(π‘₯)=βˆ’2+√π‘₯βˆ’3 where π‘₯β©Ύ3

Q14:

Find the inverse of the function 𝑓(π‘₯)=π‘₯+6π‘₯+11, where π‘₯β‰₯βˆ’3.

  • A𝑓(π‘₯)=√π‘₯βˆ’3βˆ’2
  • B𝑓(π‘₯)=√π‘₯+2+3
  • C𝑓(π‘₯)=√π‘₯βˆ’2βˆ’3
  • D𝑓(π‘₯)=βˆ’3βˆ’βˆšπ‘₯βˆ’2
  • E𝑓(π‘₯)=3βˆ’βˆšπ‘₯βˆ’2

Q15:

Find 𝑓(π‘₯) for 𝑓(π‘₯)=√π‘₯+3 and state the domain.

  • A𝑓(π‘₯)=(βˆ’π‘₯βˆ’3) for π‘₯β‰₯βˆ’3
  • B𝑓(π‘₯)=βˆ’(π‘₯βˆ’3) for π‘₯β‰₯3
  • C𝑓(π‘₯)=(π‘₯βˆ’3) for π‘₯β‰₯3
  • D𝑓(π‘₯)=(π‘₯βˆ’2) for π‘₯β‰₯2

Q16:

Find 𝑓(π‘₯) for 𝑓(π‘₯)=3+√π‘₯.

  • A𝑓(π‘₯)=π‘₯βˆ’3
  • B𝑓(π‘₯)=(π‘₯βˆ’3)
  • C𝑓(π‘₯)=βˆ’(π‘₯βˆ’3)
  • D𝑓(π‘₯)=βˆ’3βˆ’βˆšπ‘₯

Q17:

The period 𝑇, in seconds, of a simple pendulum as a function of its length 𝑙, in feet, is given by 𝑇(𝑙)=2πœ‹ο„žπ‘™32.2. Express 𝑙 as a function of 𝑇, and determine the length of a pendulum with a period of 2 seconds.

  • A𝑙=32.2𝑇2πœ‹οˆοŠ¨, 3.26 feet
  • B𝑙=ο€Ό2πœ‹π‘‡32.2, 0.15 feet
  • C𝑙=32.2𝑇2πœ‹οŠ¨, 20.5 feet
  • D𝑙=32.2𝑇2πœ‹, 10.2 feet
  • E𝑙=2πœ‹π‘‡32.2, 0.39 feet

Q18:

The figure below represents the function π‘“π‘‹β†’π‘Œ:. Find the value of 𝑓(4).

  • A10
  • B13
  • C8
  • D4

Q19:

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

  • A𝑐>7
  • B𝑐<7
  • Cany 𝑐β‰₯0
  • D𝑐<0
  • E𝑐β‰₯βˆ’7

Q20:

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?

  • Aπ‘₯>0
  • Bπ‘₯β‰€βˆ’3
  • Cπ‘₯β‰₯βˆ’3
  • Dπ‘₯β‰₯0
  • Eπ‘₯<0

What is the range of the inverse?

  • Aπ‘₯>0
  • Bπ‘₯β‰₯βˆ’3
  • Cπ‘₯β‰₯0
  • Dπ‘₯β‰€βˆ’3
  • Eπ‘₯<βˆ’3

Give a formula for the inverse.

  • A𝑓(π‘₯)=π‘₯3βˆ’3
  • B𝑓(π‘₯)=βˆ’π‘₯3+3
  • C𝑓(π‘₯)=βˆ’π‘₯3βˆ’3
  • D𝑓(π‘₯)=π‘₯3+3
  • E𝑓(π‘₯)=3π‘₯βˆ’13

Q21:

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‘₯1234𝑑6
𝑓(π‘₯)βˆ’31π‘Žβˆ’19βˆ’10114
π‘₯βˆ’31βˆ’26βˆ’19βˆ’101𝑒
𝑔(π‘₯)12𝑏𝑐56
  • Aπ‘Ž=βˆ’26, 𝑏=3, 𝑐=4, 𝑑=5, 𝑒=14
  • Bπ‘Ž=2, 𝑏=βˆ’19, 𝑐=4, 𝑑=5, 𝑒=14
  • Cπ‘Ž=2, 𝑏=βˆ’19, 𝑐=4, 𝑑=1, 𝑒=6
  • Dπ‘Ž=βˆ’26, 𝑏=βˆ’19, 𝑐=4, 𝑑=1, 𝑒=14
  • Eπ‘Ž=βˆ’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, 𝑑, given in hours by 𝑑(𝑑)=50𝑑. Find the inverse function expressing the time of travel in terms of the distance traveled. Find 𝑑(180).

  • A𝑑(𝑑)=𝑑50, 3.6 hours
  • B𝑑(𝑑)=502𝑑, 0.13889 hours
  • C𝑑(𝑑)=𝑑50, 0.2778 hours
  • D𝑑(𝑑)=2𝑑50, 7.2 hours
  • E𝑑(𝑑)=50𝑑, 0.2778 hours

Q23:

Use the table to find β„Ž(30).

π‘₯15304560
β„Ž(π‘₯)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 π‘₯βˆˆβ„βˆ’0?

  • A𝑓(π‘₯)=1π‘₯, 𝑔(π‘₯)=1π‘₯
  • B𝑓(π‘₯)=π‘₯, 𝑔(π‘₯)=βˆ’π‘₯
  • C𝑓(π‘₯)=1π‘₯, 𝑔(π‘₯)=π‘₯
  • D𝑓(π‘₯)=2π‘₯–3, 𝑔(π‘₯)=3π‘₯+2

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