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.

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

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

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

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

  • A 𝑏 = βˆ’ 4 √ 3 9
  • B 𝑏 = βˆ’ 2 √ 3 9
  • C 𝑏 = 4 √ 3 9
  • D 𝑏 = 2 √ 3 9
  • E 𝑏 = 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
  • B 1 √ 3 and βˆ’2√3
  • C √ 3 and βˆ’5√3
  • D 1 √ 3 and βˆ’βˆš39
  • E 1 √ 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.

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

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

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

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

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

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.

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

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

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

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

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

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