Worksheet: Graphs of Inverses of Functions

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In this worksheet, we will practice using a graph to find the inverse of a function and analyzing the graphs for the inverse of a function.

Q1:

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

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

  • A(b)
  • B(a)
  • C(c)

Q2:

Liam is looking for an inverse to 𝑓(𝑥)=−2−(𝑥−4). He starts with the parabola 𝑦=−2−(𝑥−4). He then reflects this in the line 𝑦=𝑥 to get the shown parabola 𝑥=−2−(𝑦−4).

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

  • A 𝑓 ( 𝑥 ) = 4 − √ − 𝑥 − 2
  • B 𝑓 ( 𝑥 ) = 4 + √ 𝑥 − 2
  • C 𝑓 ( 𝑥 ) = 4 − √ 𝑥 + 2
  • D 𝑓 ( 𝑥 ) = 4 − √ 𝑥 − 2
  • E 𝑓 ( 𝑥 ) = 4 + √ − 𝑥 − 2

Q3:

The following graph is of the function 𝑓(𝑥)=6𝑥+8𝑥+1, with its maximum at 13,9, minimum at (−3,−1), and zero at −43 marked.

Find an expression for the inverse function 𝑓 when 𝑓 is restricted to the interval 𝑥≥13.

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

What is the domain of 𝑓 in this case?

  • A − 1 ≤ 𝑥 < 9
  • B 𝑥 ≥ 1 3
  • C − 1 < 𝑥 ≤ 9
  • D 0 < 𝑥 ≤ 9
  • E 0 ≤ 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 = − 4 √ 3 9
  • B 𝑏 = − 2 √ 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 √ 3 and −1√3
  • B 1 √ 3 and −2√3
  • C √ 3 and −5√3
  • D 1 √ 3 and −√39
  • E 1 √ 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 45,45.

Determine the value of 𝑏.

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

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

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

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

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

Q6:

The following graph is of the function 𝑓(𝑥)=6𝑥+8𝑥+1, with its maximum at 13,9, minimum at (−3,−1), and zero at −43 marked.

Find an expression for the inverse function 𝑓 when 𝑓 is restricted to the interval −3≤𝑥<−43.

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

What is the domain of 𝑓 in this case?

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

Q7:

The following graph is of the function 𝑓(𝑥)=6𝑥+8𝑥+1, with its maximum at 13,9, minimum at (−3,−1), and zero at −43 labeled.

Find an expression for the inverse function 𝑓 when 𝑓 is restricted to the interval −43<𝑥≤13.

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

What is the domain of 𝑓 in this case?

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

  • ANo
  • BYes

Q10:

By sketching the graphs of 𝑓(𝑥)=3𝑥−1 and 𝑔(𝑥)=√𝑥+13 for 𝑥≥0, determine whether they are inverse functions.

  • AThey are not inverse functions.
  • BThey are inverse functions.

Q11:

By sketching the graphs of 𝑓(𝑥)=2𝑥 and 𝑔(𝑥)=𝑥2 for 𝑥≥0, determine whether they are inverse functions.

  • AThey are not inverse functions.
  • BThey are inverse functions.

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