Worksheet: Finding the Inverse of a Function from Its Graph

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In this worksheet, we will practice using a graph to find the inverse of a function and will explore the symmetry between the graph of a function and that of its inverse.

Q1:

The following is the graph of 𝑓 ( 𝑥 ) = 2 𝑥 − 1 .

Which one is the graph of the inverse function 𝑓 ( 𝑥 ) − 1 ?

  • A (c)
  • B (b)
  • C (a)

Q2:

Amir is looking for an inverse to . He starts with the parabola . He then reflects this in the line to get the shown parabola .

Complete Amir’s work by determining the inverse whose graph is the given solid curve.

  • A
  • B
  • C
  • D
  • E

Q3:

The following graph is of the function 𝑓 ( 𝑥 ) = 6 𝑥 + 8 𝑥 + 1 2 , with its maximum at 1 3 , 9 , minimum at ( − 3 , − 1 ) , and zero at − 4 3 marked.

Find an expression for the inverse function 𝑓 − 1 when 𝑓 is restricted to the interval 𝑥 ≥ 1 3 .

  • A 𝑓 ( 𝑥 ) = 3 − √ − 𝑥 + 8 𝑥 + 9 𝑥 − 1 2
  • B 𝑓 ( 𝑥 ) = 𝑥 + 1 6 𝑥 + 8 − 1 2
  • C 𝑓 ( 𝑥 ) = 3 + √ − 𝑥 + 8 𝑥 + 9 − 1 2
  • D 𝑓 ( 𝑥 ) = 3 + √ − 𝑥 + 8 𝑥 + 9 𝑥 − 1 2
  • E 𝑓 ( 𝑥 ) = 3 − √ − 𝑥 + 8 𝑥 + 9 − 1 2

What is the domain of 𝑓 − 1 in this case?

  • A 0 < 𝑥 ≤ 9
  • B − 1 ≤ 𝑥 < 9
  • C 0 ≤ 𝑥 ≤ 9
  • D 𝑥 ≥ 1 3
  • E − 1 < 𝑥 ≤ 9

Q4:

Consider the two following figures.

The first figure shows the graph of 𝑓 ( 𝑥 ) = 𝑥 and a tangent to the graph with gradient 1. This tangent meets the graph at a point with 𝑥 -coordinate 1 √ 3 .

The second figure shows the graphs of 𝑔 ( 𝑥 ) = 𝑥 + 𝑏 and its inverse 𝑔 ( 𝑥 ) = ( 𝑥 − 𝑏 ) . The graphs cross in the third quadrant and touch in the first quadrant.

What is the value of 𝑏 ?

  • A 𝑏 = − 2 √ 3 9
  • B 𝑏 = 4 √ 3 9
  • C 𝑏 = − 4 √ 3 9
  • D 𝑏 = 2 √ 3 9
  • E 𝑏 = − 1 √ 3

What are the 𝑥 -coordinates of the two points of intersection of the graphs in the second figure?

  • A 1 √ 3 and − 2 √ 3
  • B 1 √ 3 and − 5 √ 3
  • C 1 √ 3 and − √ 3 9
  • D √ 3 and − 1 √ 3
  • E √ 3 and − 5 √ 3

Q5:

The graphs of 𝑓 ( 𝑥 ) = 𝑥 + 𝑏 and its inverse 𝑓 ( 𝑥 ) intersect at three points, one of which is 4 5 , 4 5 .

Determine the value of 𝑏 .

  • A 6 4 1 2 5
  • B 1 6 4 1 2 5
  • C 4 5
  • D 3 6 1 2 5
  • E − 4 5

Find the 𝑥 -coordinate of the point 𝐴 marked on the figure.

  • A 3 6 1 2 5
  • B 4 5
  • C 6 4 1 2 5
  • D 1 6 4 1 2 5
  • E − 4 5

Find the 𝑥 -coordinate of the point 𝐵 marked on the figure.

  • A 8 2 5
  • B − 2 − √ 1 3 5
  • C √ 1 3 − 2 5
  • D 3 6 1 2 5
  • E 1 3

Q6:

The following graph is of the function , with its maximum at , minimum at , and zero at marked.

Find an expression for the inverse function when is restricted to the interval .

  • A
  • B
  • C
  • D
  • E

What is the domain of in this case?

  • A
  • B
  • C
  • D
  • E

Q7:

The following graph is of the function 𝑓 ( 𝑥 ) = 6 𝑥 + 8 𝑥 + 1 2 , with its maximum at 1 3 , 9 , minimum at ( − 3 , − 1 ) , and zero at − 4 3 labeled.

Find an expression for the inverse function 𝑓 − 1 when 𝑓 is restricted to the interval − 4 3 < 𝑥 ≤ 1 3 .

  • A 𝑓 ( 𝑥 ) = 3 + √ − 𝑥 + 8 𝑥 + 9 𝑥 − 1 2
  • B 𝑓 ( 𝑥 ) = 𝑥 + 1 6 𝑥 + 8 − 1 2
  • C 𝑓 ( 𝑥 ) = 3 + √ − 𝑥 + 8 𝑥 + 9 − 1 2
  • D 𝑓 ( 𝑥 ) = 3 − √ − 𝑥 + 8 𝑥 + 9 𝑥 − 1 2
  • E 𝑓 ( 𝑥 ) = 3 − √ − 𝑥 + 8 𝑥 + 9 − 1 2

What is the domain of 𝑓 − 1 in this case?

  • A 0 < 𝑥 ≤ 9
  • B − 1 ≤ 𝑥 ≤ 9
  • C − 1 < 𝑥 ≤ 9
  • D − 4 3 < 𝑥 ≤ 1 3
  • E 0 ≤ 𝑥 ≤ 9

Q8:

Determine whether the inverse of the represented function is a function or not.

  • Anot a function
  • Ba function

Q9:

In the given figure, the green points represent the function 𝑓 ( 𝑥 ) . Do the blue points represent 𝑓 ( 𝑥 ) − 1 ?

  • AYes
  • BNo

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