Worksheet: Graphs of Inverses of Functions

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(π‘₯)=2π‘₯βˆ’1.

Which one is the graph of the inverse function 𝑓(π‘₯)?

  • A(b)
  • B(a)
  • C(c)

Q2:

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

Complete Liam’s work by determining the inverse π‘“οŠ±οŠ§ whose graph is the given solid curve.

  • A𝑓(π‘₯)=4βˆ’βˆšβˆ’π‘₯βˆ’2
  • B𝑓(π‘₯)=4+√π‘₯βˆ’2
  • C𝑓(π‘₯)=4βˆ’βˆšπ‘₯+2
  • D𝑓(π‘₯)=4βˆ’βˆšπ‘₯βˆ’2
  • E𝑓(π‘₯)=4+βˆšβˆ’π‘₯βˆ’2

Q3:

The following graph is of the function 𝑓(π‘₯)=6π‘₯+8π‘₯+1, with its maximum at ο€Ό13,9, minimum at (βˆ’3,βˆ’1), and zero at βˆ’43 marked.

Find an expression for the inverse function π‘“οŠ±οŠ§ when 𝑓 is restricted to the interval π‘₯β‰₯13.

  • A𝑓(π‘₯)=3βˆ’βˆšβˆ’π‘₯+8π‘₯+9
  • B𝑓(π‘₯)=3βˆ’βˆšβˆ’π‘₯+8π‘₯+9π‘₯
  • C𝑓(π‘₯)=3+βˆšβˆ’π‘₯+8π‘₯+9π‘₯
  • D𝑓(π‘₯)=3+βˆšβˆ’π‘₯+8π‘₯+9
  • E𝑓(π‘₯)=π‘₯+16π‘₯+8

What is the domain of π‘“οŠ±οŠ§ in this case?

  • Aβˆ’1≀π‘₯<9
  • Bπ‘₯β‰₯13
  • Cβˆ’1<π‘₯≀9
  • D0<π‘₯≀9
  • E0≀π‘₯≀9

Q4:

Consider the two following figures.

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

The second figure shows the graphs of 𝑔(π‘₯)=π‘₯+π‘οŠ© and its inverse 𝑔(π‘₯)=(π‘₯βˆ’π‘). The graphs cross in the third quadrant and touch in the first quadrant.

What is the value of 𝑏?

  • A𝑏=βˆ’4√39
  • B𝑏=βˆ’2√39
  • C𝑏=4√39
  • D𝑏=2√39
  • E𝑏=βˆ’1√3

What are the π‘₯-coordinates of the two points of intersection of the graphs in the second figure?

  • A√3 and βˆ’1√3
  • B1√3 and βˆ’2√3
  • C√3 and βˆ’5√3
  • D1√3 and βˆ’βˆš39
  • E1√3 and βˆ’5√3

Q5:

The graphs of 𝑓(π‘₯)=π‘₯+π‘οŠ© and its inverse 𝑓(π‘₯) intersect at three points, one of which is ο€Ό45,45.

Determine the value of 𝑏.

  • A64125
  • B45
  • C36125
  • D164125
  • Eβˆ’45

Find the π‘₯-coordinate of the point 𝐴 marked on the figure.

  • A45
  • B164125
  • Cβˆ’45
  • D36125
  • E64125

Find the π‘₯-coordinate of the point 𝐡 marked on the figure.

  • Aβˆ’2βˆ’βˆš135
  • B825
  • C36125
  • D13
  • E√13βˆ’25

Q6:

The following graph is of the function 𝑓(π‘₯)=6π‘₯+8π‘₯+1, with its maximum at ο€Ό13,9, minimum at (βˆ’3,βˆ’1), and zero at βˆ’43 marked.

Find an expression for the inverse function π‘“οŠ±οŠ§ when 𝑓 is restricted to the interval βˆ’3≀π‘₯<βˆ’43.

  • A𝑓(π‘₯)=3βˆ’βˆšβˆ’π‘₯+8π‘₯+9π‘₯
  • B𝑓(π‘₯)=3+βˆšβˆ’π‘₯+8π‘₯+9π‘₯
  • C𝑓(π‘₯)=π‘₯+16π‘₯+8
  • D𝑓(π‘₯)=3+βˆšβˆ’π‘₯+8π‘₯+9
  • E𝑓(π‘₯)=3βˆ’βˆšβˆ’π‘₯+8π‘₯+9

What is the domain of π‘“οŠ±οŠ§ in this case?

  • Aβˆ’1<π‘₯≀9
  • Bβˆ’3≀π‘₯<βˆ’43
  • Cβˆ’1≀π‘₯<0
  • Dβˆ’1<π‘₯≀0
  • Eβˆ’1≀π‘₯<9

Q7:

The following graph is of the function 𝑓(π‘₯)=6π‘₯+8π‘₯+1, with its maximum at ο€Ό13,9, minimum at (βˆ’3,βˆ’1), and zero at βˆ’43 labeled.

Find an expression for the inverse function π‘“οŠ±οŠ§ when 𝑓 is restricted to the interval βˆ’43<π‘₯≀13.

  • A𝑓(π‘₯)=3+βˆšβˆ’π‘₯+8π‘₯+9π‘₯
  • B𝑓(π‘₯)=3βˆ’βˆšβˆ’π‘₯+8π‘₯+9
  • C𝑓(π‘₯)=3+βˆšβˆ’π‘₯+8π‘₯+9
  • D𝑓(π‘₯)=π‘₯+16π‘₯+8
  • E𝑓(π‘₯)=3βˆ’βˆšβˆ’π‘₯+8π‘₯+9π‘₯

What is the domain of π‘“οŠ±οŠ§ in this case?

  • Aβˆ’1≀π‘₯≀9
  • B0≀π‘₯≀9
  • Cβˆ’43<π‘₯≀13
  • D0<π‘₯≀9
  • Eβˆ’1<π‘₯≀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(π‘₯). Do the blue points represent 𝑓(π‘₯)?

  • ANo
  • BYes

Q10:

By sketching the graphs of 𝑓(π‘₯)=3π‘₯βˆ’1 and 𝑔(π‘₯)=√π‘₯+13 for π‘₯β‰₯0, determine whether they are inverse functions.

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

Q11:

By sketching the graphs of 𝑓(π‘₯)=2π‘₯ and 𝑔(π‘₯)=ο„žπ‘₯2 for π‘₯β‰₯0, determine whether they are inverse functions.

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

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