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Lesson: Using the Graph of a Function to Find Its Inverse

Sample Question Videos

Worksheet • 9 Questions • 1 Video

Q1:

Liam is looking for an inverse to 𝑓 ( π‘₯ ) = βˆ’ 2 βˆ’ ( π‘₯ βˆ’ 4 ) 2 . He starts with the parabola 𝑦 = βˆ’ 2 βˆ’ ( π‘₯ βˆ’ 4 ) 2 . He then reflects this in the line 𝑦 = π‘₯ to get the shown parabola π‘₯ = βˆ’ 2 βˆ’ ( 𝑦 βˆ’ 4 ) 2 .

Complete Liam’s work by determining the inverse 𝑓 βˆ’ 1 whose graph is the given solid curve.

  • A 𝑓 ( π‘₯ ) = 4 βˆ’ √ βˆ’ π‘₯ βˆ’ 2 βˆ’ 1
  • B 𝑓 ( π‘₯ ) = 4 + √ π‘₯ βˆ’ 2 βˆ’ 1
  • C 𝑓 ( π‘₯ ) = 4 + √ βˆ’ π‘₯ βˆ’ 2 βˆ’ 1
  • D 𝑓 ( π‘₯ ) = 4 βˆ’ √ π‘₯ + 2 βˆ’ 1
  • E 𝑓 ( π‘₯ ) = 4 βˆ’ √ π‘₯ βˆ’ 2 βˆ’ 1

Q2:

The following graph is of the function 𝑓 ( π‘₯ ) = 6 π‘₯ + 8 π‘₯ + 1 2 , with its maximum at ο€Ό 1 3 , 9  , minimum at ( βˆ’ 3 , βˆ’ 1 ) , and zero at βˆ’ 4 3 labeled.

Find an expression for the inverse function 𝑓 βˆ’ 1 when 𝑓 is restricted to the interval βˆ’ 4 3 < π‘₯ ≀ 1 3 .

  • A 𝑓 ( π‘₯ ) = 3 βˆ’ √ βˆ’ π‘₯ + 8 π‘₯ + 9 π‘₯ βˆ’ 1 2
  • B 𝑓 ( π‘₯ ) = 3 βˆ’ √ βˆ’ π‘₯ + 8 π‘₯ + 9 βˆ’ 1 2
  • C 𝑓 ( π‘₯ ) = π‘₯ + 1 6 π‘₯ + 8 βˆ’ 1 2
  • D 𝑓 ( π‘₯ ) = 3 + √ βˆ’ π‘₯ + 8 π‘₯ + 9 βˆ’ 1 2
  • E 𝑓 ( π‘₯ ) = 3 + √ βˆ’ π‘₯ + 8 π‘₯ + 9 π‘₯ βˆ’ 1 2

What is the domain of 𝑓 βˆ’ 1 in this case?

  • A 0 < π‘₯ ≀ 9
  • B 0 ≀ π‘₯ ≀ 9
  • C βˆ’ 4 3 < π‘₯ ≀ 1 3
  • D βˆ’ 1 ≀ π‘₯ ≀ 9
  • E βˆ’ 1 < π‘₯ ≀ 9

Q3:

The following graph is of the function 𝑓 ( π‘₯ ) = 6 π‘₯ + 8 π‘₯ + 1 2 , with its maximum at ο€Ό 1 3 , 9  , minimum at ( βˆ’ 3 , βˆ’ 1 ) , and zero at βˆ’ 4 3 marked.

Find an expression for the inverse function 𝑓 βˆ’ 1 when 𝑓 is restricted to the interval π‘₯ β‰₯ 1 3 .

  • A 𝑓 ( π‘₯ ) = 3 + √ βˆ’ π‘₯ + 8 π‘₯ + 9 π‘₯ βˆ’ 1 2
  • B 𝑓 ( π‘₯ ) = 3 βˆ’ √ βˆ’ π‘₯ + 8 π‘₯ + 9 βˆ’ 1 2
  • C 𝑓 ( π‘₯ ) = π‘₯ + 1 6 π‘₯ + 8 βˆ’ 1 2
  • D 𝑓 ( π‘₯ ) = 3 + √ βˆ’ π‘₯ + 8 π‘₯ + 9 βˆ’ 1 2
  • E 𝑓 ( π‘₯ ) = 3 βˆ’ √ βˆ’ π‘₯ + 8 π‘₯ + 9 π‘₯ βˆ’ 1 2

What is the domain of 𝑓 βˆ’ 1 in this case?

  • A 0 < π‘₯ ≀ 9
  • B βˆ’ 1 < π‘₯ ≀ 9
  • C π‘₯ β‰₯ 1 3
  • D βˆ’ 1 ≀ π‘₯ < 9
  • E 0 ≀ π‘₯ ≀ 9
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