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1. saudjanaksingh@gmail.com
6/19/2020 JANAK SINGH SAUD 1

2. Review: Function
• https://www.geogebra.org/m/mzxvxe44
INPUT Num.
OUTPUT Num.
(INPUT, OUTPUT)
(3, 9)
Domain
Range
6/19/2020 JANAK SINGH SAUD 2
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3. https://www.geogebra.org/m/vP9qTDF4
6/19/2020 JANAK SINGH SAUD 3
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4. 6/19/2020 JANAK SINGH SAUD 4

5. Many to one-onto function
6/19/2020 JANAK SINGH SAUD 5
1
2
3
A B
5
After interchanging domain and range
B
5
1
2
3
A
It is not a
function
Many to one-into function
1
2
3
A B
3
4
5
After interchanging domain and range
B
3
4
5
1
2
3
A
It is not a
function

6. One to one- into function
6/19/2020 JANAK SINGH SAUD 6
1
2
3
A B
1
4
9
16
After interchanging domain and range
B
1
2
9
16
1
2
3
A
It is not a
function
One to one- onto function
1
2
3
A B
1
4
9
After interchanging domain and range
B
1
4
9
1
2
3
A
It is function

7. 6/19/2020 JANAK SINGH SAUD 7
One to one- onto function
1
2
3
A B
1
4
9
B
1
4
9
1
2
3
Af
f-1
Domain
Range Domain
Range
Range of f = Domain of f-1
1
4
9
1
4
9
Domain of f = Range of f -1
1
2
3
1
2
3

8. 6/19/2020 JANAK SINGH SAUD 8
Range
Domain
Function (f)
Inverse (f -1 )
Domain
Range
https://tasyaluftwascher.blogspot.com/2017/01/newest-for-images-
drawing-cute-water.htmlSource

9. Conclusion
:
6/19/2020 JANAK SINGH SAUD 9
A B B A
f f-1
1
4
9
1
4
9
1
2
3
1
2
3
f(x) =y f(x) =yxx

10. Find the inverse of the function f(x) = x – 5 and show in arrow diagram also.
• Here, y = f(x) = x – 5
• y = x – 5
• Interchanging domain (x) and range (y), we have
• x = y - 5
• x + 5 = y
• y = x + 5
∴ f-1(x) = x + 5
6/19/2020 JANAK SINGH SAUD 10
EXAMPLE 1
1
2
3
A B
- 4
-3
-2
B
-4
-3
-2
1
2
3
A
f
f-1 • f(1) = 1 – 5 = - 4
• f(2) = 2 – 5 = - 3
• f(3) = 3 – 5 = - 2
• f – 1(- 4) = - 4 + 5 = 1
• f – 1( - 3) = - 3 + 5 = 2
• f – 1(- 2) = - 2 + 5 = 3
https://www.geogebra.org/m/mzxvxe44

11. Find the inverse of the function f(x) = 𝑥,
3𝑥−1
𝑥+1
: 𝑥 ∈ 𝑅
• Here, f(x) = y =
𝟑𝒙−𝟏
𝒙+𝟏
• y =
𝟑 𝒙 −𝟏
𝒙 +𝟏
• Interchanging domain (x) And
range (y), we have
• x =
𝟑 𝒚 −𝟏
𝒚 +𝟏
•
𝒙
𝟏
=
𝟑𝒚−𝟏
𝒚+𝟏
6/19/2020 JANAK SINGH SAUD 11
EXAMPLE 2
• 3y – 1 = xy + x
• 3y – xy = x + 1
• (3 – x) y = x + 1
• Y =
𝒙+𝟏
𝟑 −𝒙
• ∴ f -1(x) =
𝒙+𝟏
𝟑 −𝒙

12. If f(x+ 3) = 5x + 4, then find f -1(x)
• Here, f(x +3 ) = 5x+ 4
• Let x + 3 = k. Then
• x = k – 3
• ∴ f(k) = 5(k – 3) + 4
• f(k) = 5k – 15 + 4
• f(k) = 5k – 11
• ∴ f(x) = 5x - 11
6/19/2020 JANAK SINGH SAUD 12
EXAMPLE 3
• Again, y = f(x) = 5x – 11
• y = 5 x – 11
• Interchanging domain (x) and
range (y), we get
• x = 5 y – 11
• x + 11 = 5y
•
𝒙+𝟏𝟏
𝟓
= y
• y =
𝒙+𝟏𝟏
𝟓
• ∴ f -1(x) =
𝒙+𝟏𝟏
𝟓
Replace k = x

13. Home work
• Find the inverse of the following functions and show in the arrow
diagrams
1. f(x) = 5 – 7x
2. g(x) =
3𝑥+1
4
3. h(x) = {(1, 1), (2, 4), (3, 9), (4, 16)}.
4. f(x) = {(1, 3), (2, 3),(3, 5),(4, 6),(5, 7}. if exists.
5. if f(x) =
𝑥+1
𝑥 −1
find f-1(2)
6/19/2020 JANAK SINGH SAUD 13

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